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The general rule is that any damage is a roll

As you have noted, a critical hit permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that damage is defined as a die or dice as a general rule. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'.

Some damage effects (like oil or an unarmed attack) specify that they do a certain amount of base damage, which is not rolled. This is a specific exception to the general rule that damage is rolled on a damage die or dice. The sprite longsword itself includes no such rule (but see below). However, monster damage has its own specific rules as presented in the Monster Manual (emphasis in the original):

Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation. You have the option of taking average damage or rolling the damage; for this reason, both the average damage and the die expression are presented.

For the sprite, the damage is presented as:

Hit: 1 slashing damage.

Since there is only one damage presented ("1") and with the expectation that "both the average damage and the die expression are presented", "1" must therefor represent both the average damage and the die roll. Thus the roll is a 1d1, whose only outcome is 1 damage, which is also the average of the roll. However, on a critical, the damage dice would be doubled to 2d1, or 2.

Roll20 supports this interpretation

In roll20, the damage listing for a sprite longsword is presented as:

Hit: 1 (1d1) slashing damage

Here, the damage roll of 1d1 is made explicit and, when a sprite rolls a critical hit, the damage is presented as 1 + 1, as predicted if the base damage of 1 represented a roll.

It should be noted that roll20 offers an unofficial interpretation of RAW. For example, the roll20 descriptions of spells explicitly list spell targets, while in the PHB these are generally implicit. That roll20 chooses to represent sprite longsword damage as a roll is interesting support, but not an official interpretation.

The general rule is that any damage is a roll

As you have noted, a critical hit permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that damage is defined as a die or dice as a general rule. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'.

Some damage effects (like oil or an unarmed attack) specify that they do a certain amount of base damage, which is not rolled. This is a specific exception to the general rule that damage is rolled on a damage die or dice. The sprite longsword itself includes no such rule (but see below). However, monster damage has its own specific rules as presented in the Monster Manual (emphasis in the original):

Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation. You have the option of taking average damage or rolling the damage; for this reason, both the average damage and the die expression are presented.

For the sprite, the damage is presented as:

Hit: 1 slashing damage.

Since there is only one damage presented ("1") and with the expectation that "both the average damage and the die expression are presented", "1" must therefor represent both the average damage and the die roll. Thus the roll is a 1d1, whose only outcome is 1 damage, which is also the average of the roll. However, on a critical, the damage dice would be doubled to 2d1, or 2.

Roll20 supports this interpretation

In roll20, the damage listing for a sprite longsword is presented as:

Hit: 1 (1d1) slashing damage

Here, the damage roll of 1d1 is made explicit and, when a sprite rolls a critical hit, the damage is shown as 1 + 1 in the text bar, as predicted if the base damage of 1 represented a roll.

It should be noted that roll20 offers an unofficial interpretation of RAW. For example, the roll20 descriptions of spells explicitly list spell targets, while in the PHB these are generally implicit. That roll20 chooses to represent sprite longsword damage as a roll is interesting support, but not an official interpretation.

The general rule is that any damage is a roll

As you have noted, a critical hit permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that damage is defined as a die or dice as a general rule. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'.

Some damage effects (like oil or an unarmed attack) specify that they do a certain amount of base damage, which is not rolled. This is a specific exception to the general rule that damage is rolled on a damage die or dice. The sprite longsword itself includes no such rule (but see below). However, monster damage has its own specific rules as presented in the Monster Manual (emphasis in the original):

Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation. You have the option of taking average damage or rolling the damage; for this reason, both the average damage and the die expression are presented.

For the sprite, the damage is presented as:

Hit: 1 slashing damage.

Since there is only one damage presented ("1") and with the rule that "both the average damage and the die expression are presented", "1" must therefor represent both the average damage and the die roll. Thus the roll is a 1d1, whose only outcome is 1 damage, which is also the average of the roll. However, on a critical, the damage dice would be doubled to 2d1, or 2.

Roll20 supports this interpretation

In roll20, the damage listing for a sprite longsword is presented as:

Hit: 1 (1d1) slashing damage

Here, the damage roll of 1d1 is made explicit and, when a sprite rolls a critical hit, the damage is presented as 1 + 1, as predicted if the base damage of 1 represented a roll.

It should be noted that roll20 offers an unofficial interpretation of RAW. For example, the roll20 descriptions of spells explicitly list spell targets, while in the PHB these are generally implicit. That roll20 chooses to represent sprite longsword damage as a roll is interesting support, but not an official interpretation.

The general rule is that any damage is a roll

As you have noted, a critical hit permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that damage is defined as a die or dice as a general rule. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'.

Some damage effects (like oil or an unarmed attack) specify that they do a certain amount of base damage, which is not rolled. This is a specific exception to the general rule that damage is rolled on a damage die or dice. The sprite longsword itself includes no such rule (but see below). However, monster damage has its own specific rules as presented in the Monster Manual (emphasis in the original):

Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation. You have the option of taking average damage or rolling the damage; for this reason, both the average damage and the die expression are presented.

For the sprite, the damage is presented as:

Hit: 1 slashing damage.

Since there is only one damage presented ("1") and with the expectation that "both the average damage and the die expression are presented", "1" must therefor represent both the average damage and the die roll. Thus the roll is a 1d1, whose only outcome is 1 damage, which is also the average of the roll. However, on a critical, the damage dice would be doubled to 2d1, or 2.

Roll20 supports this interpretation

In roll20, the damage listing for a sprite longsword is presented as:

Hit: 1 (1d1) slashing damage

Here, the damage roll of 1d1 is made explicit and, when a sprite rolls a critical hit, the damage is presented as 1 + 1, as predicted if the base damage of 1 represented a roll.

It should be noted that roll20 offers an unofficial interpretation of RAW. For example, the roll20 descriptions of spells explicitly list spell targets, while in the PHB these are generally implicit. That roll20 chooses to represent sprite longsword damage as a roll is interesting support, but not an official interpretation.

RAW any damage is a roll

As you have noted, a critical permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that all damage is defined as a die or dice. From the PHB on Damage Rolls (emphasis mine):

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is always a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'. For a hit with a sprite-wielded longsword that does a base of 1 damage, there is no RAW way to apply this amount as damage unless the value of 1 represents a roll.

For damage that is specified as a flat amount x, the damage die must be of the form xd1. A sprite longsword (or blowgun needle, or unarmed strike) which does 1 point before the Strength modifier is thus a 1d1, whose only outcome is 1 damage. On a critical the damage dice would be doubled to 2d1, or 2.

At least, this is what we can conclude from the language in the PHB. Is this what is intended? Most likely not, and see below. But it is what is written.

RAI flat values are not rolls

As Jeremy Crawford tweeted,

An unarmed strike deals 1 + Str. modifier damage, even on a critical hit.

The intention is that an unrolled damage amount is not increased on a critical.

Roll20 treats flat damage values as rolls

If you use roll20, you may notice that a flat value is treated as a damage roll. For example, if your character is equipped with a blowgun, the needle will do 1 point of damage (plus your Dex mod if you are proficient). But on a critical hit, it will display the damage as 1 + 1.

That the calculation of damage for a critical hit is applied in this manner in roll20 says nothing about RAW or RAI for the rule, but it is worth noting in case this application differs from the ruling at your table.

The general rule is that any damage is a roll

As you have noted, a critical hit permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that damage is defined as a die or dice as a general rule. From the PHB on Damage Rolls (emphasis mine):

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'.

Some damage effects (like oil or an unarmed attack) specify that they do a certain amount of base damage, which is not rolled. This is a specific exception to the general rule that damage is rolled on a damage die or dice. The sprite longsword itself includes no such rule (but see below). However, monster damage has its own specific rules as presented in the Monster Manual (emphasis in the original):

Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the "Hit" notation. You have the option of taking average damage or rolling the damage; for this reason, both the average damage and the die expression are presented.

For the sprite, the damage is presented as:

Hit: 1 slashing damage.

Since there is only one damage presented ("1") and with the rule that "both the average damage and the die expression are presented", "1" must therefor represent both the average damage and the die roll. Thus the roll is a 1d1, whose only outcome is 1 damage, which is also the average of the roll. However, on a critical, the damage dice would be doubled to 2d1, or 2.

Roll20 supports this interpretation

In roll20, the damage listing for a sprite longsword is presented as:

Hit: 1 (1d1) slashing damage

Here, the damage roll of 1d1 is made explicit and, when a sprite rolls a critical hit, the damage is presented as 1 + 1, as predicted if the base damage of 1 represented a roll.

It should be noted that roll20 offers an unofficial interpretation of RAW. For example, the roll20 descriptions of spells explicitly list spell targets, while in the PHB these are generally implicit. That roll20 chooses to represent sprite longsword damage as a roll is interesting support, but not an official interpretation.