Ranged Weapons Use Dex
Chapter 9 (Combat) has the rules for attacks, which say:
Ability Modifier. The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity.
The Sage Advise Compendium (v2.7, 2021, p.12) has a question, How do I know which ability modifier to use with an attack roll and its damage roll? which restates the same thing. Further, in the question What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon? (p.13) it says:
It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon...Here’s a bit of wording minutia: we would write “melee-weapon attack” (with a hyphen) if we meant an attack with a melee weapon.
Since you are throwing the crossbow, you are making a ranged attack, and thus you would use Dex. It doesn't matter whether the crossbow itself is a ranged weapon, melee weapon, or improvised weapon; since the attack is ranged, you use Dex.
As examples of specific over general, a few things can change this default expectation, but they don't apply in this case. The Finesse weapon property allows one to choose Str or Dex, but since the crossbow is not a finesse weapon, this does not apply.
The Thrown weapon property says that:
If the weapon is a melee weapon, you use the same ability modifier for that attack roll and damage roll that you would use for a melee attack with the weapon.
The crossbow is now a thrown weapon, since it was given that property by the elemental cleaver feature. However, that same feature did not turn it into a melee weapon. It remains a ranged weapon, and thus the 'damage as a melee weapon would' clause of the thrown weapon property does not apply, either.
More than one answer here states that by throwing it, the crossbow has become an improvised weapon. Even if this were true1, improvised weapons by themselves don't have any special rules about what ability score modifiers to use; they simply revert to the default rule listed above; it is a ranged attack, therefor we use Dex.
But, does that make sense?
Realistically, hucking the butt-end of an actual crossbow at someone's head should be a Strength-based improvised weapon.
But that is not what is happening here. The elemental cleaver class feature takes what is otherwise a normal weapon and converts it into a source of elemental damage:
you can infuse one weapon of your choice that you are holding with one of the following damage types: acid, cold, fire, thunder, or lightning. While you wield the infused weapon during your rage, the weapon’s damage type changes to the chosen type, it deals an extra 1d6 damage of the chosen type when it hits, and it gains the thrown property, with a normal range of 20 feet and a long range of 60 feet. If you throw the weapon, it reappears in your hand the instant after it hits or misses a target.
The fact that it does an elemental damage type, does extra damage beyond the base of the weapon, and reappears in your hand all indicate that we are not dealing with a normal crossbow treated as an improvised weapon. You are not hurling a wooden crossbow, you are throwing a thunderbolt, or launching an acid axe, and it is your accuracy that matters, not your force. For you, the weapon is an instrument of "primordial might", even if it would just be a crossbow in the hands of someone else:
The infused weapon’s benefits are suppressed while a creature other than you wields it.
Your DM may rule differently if they find RAW not useful here
The whole flavor of the subclass is that when you rage, you are converted into a thunderbolt-hurling titan. However, if that narrative is not something that resonates with your DM, if they decide that throwing a crossbow at someone's head while you are raging, even a crossbow made of acid, should be based on your STR, not your DEX, then they are empowered to make that change. Or, perhaps they agree with @enkryptor, and think that the feature itself should have been restricted to melee weapons.
As long as they are making house rules, I would suggest that they also change the thrown crossbow's ammunition property, which has not been removed by your Elemental cleaver. RAW, even though you are throwing the crossbow itself, you would need at least one bolt on hand to do it, and every time you threw the crossbow you would somehow have to expend a bolt.
1 By definition, an improvised weapon deals only d4 damage. A weapon infused with elemental cleaver deals the full base damage of the weapon plus d6 more.
An improvised weapon deals damage of the type the DM assigns, appropriate to the object. As a power of the elemental cleaver feature, the player chooses the elemental damage type of the infused weapon.
An improvised weapon can be a ranged weapon used to make a melee attack, or a thrown melee weapon that does not have the thrown property. In this case, a crossbow, the infused weapon is a ranged weapon that is thrown and does have the thrown property - which is the same as a normal dart.