Can spontaneous spellcasting archetypes swap spells?

Question

Assume a martial character with no spellcasting takes an archetype dedication feat at level 2 that gives them access to spontaneous spellcasting (bard, sorcerer, oracle). Then the character takes the subsequent level 4 feat that gives them basic spellcasting. While adding spells to the repertoire is described in the linked rules, swapping spells is not.

The full class spontaneous spellcasters have a feature called "Swapping Spells In Your Repertoire" in their class description and can therefore swap spells at level up or during downtime.

I cannot find an analogous rule for spellcasting archetypes. Are they forever stuck with their spell choices?

Note that the "Learn a Spell" exploration action only gives spontaneous casters the option to "[...] can select [the spell] when you add or swap spells".

Answer 1

Yes, but maybe only with retraining

The rules for Retraining indicates that it is possible to choose to do so...

Retraining offers a way to alter some of your character choices, which is helpful when you want to take your character in a new direction or change decisions that didn’t meet your expectations. You can retrain feats, skills, and some selectable class features.
[...]
You can change a class feature that required a choice, making a different choice instead. Some, like changing a spell in your spell repertoire, take a week. The GM will tell you how long it takes to retrain larger choices like a druid order or a wizard school—always at least a month.

The Archetype gives you a spell repertoire, so it stands to reason that guidelines for retraining still apply, even if you don't have Sorcerer Spell Repertoire class feature itself.

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I'd go a step further

I would follow all of the rules of the spell repertoire class feature for the archetypes that indicate "You gain a spell repertoire..."; all of it's abilities and limitations, except modified by the spellcasting archetype rules limiting spell slots and spells known differently than the original class.

Answer 2

No, spontaneous spellcasting archetypes cannot swap spells

The ability to swap spells is not part of the Spellcasting feature for the class that the archetype emulates, it is part of the Spell Repertoire feature.

Let's look at an archetype to see how it works. The sorcerer archetype (as an example for a sponteanous spellcasting archetype) says

You cast spells like a sorcerer. You gain access to the Cast a Spell activity. You gain a spell repertoire with two common cantrips from the spell list associated with your bloodline, from the spells granted by your bloodline, or any other cantrips of that tradition you learn or discover.

The archetype says it casts spells like a sorcerer, so it gains access to the sorcerer's Sorcerer Spellcasting feature. It however does not say that it learns spells like a sorcerer, so you do not gain access to the sorcerer's Spell Reportoire feature that includes the ability to swap. Instead it explicitly describes the nature of the spell repertoire you gain.

This is also not changed by the various spellcasting feats. The Basic Sorcerer Spellcasting says

You gain the basic spellcasting benefits. Each time you gain a spell slot of a new level from the sorcerer archetype, add a spell of the appropriate spell level to your repertoire

and the Basic spellcasting benefit for your repertoire is

these feats grant a 1st-level spell slot. At 6th level, they grant you a 2nd-level spell slot, and if you have a spell repertoire, you can select one spell from your repertoire as a signature spell. At 8th level, they grant you a 3rd-level spell slot

so for each of these spell slots you get, you may add one spell of appropriate level to your repertoire. Expert and Master spellcasting for the sorcerer just refer back to the standard Expert and Master spellcasting to expand this, for Expert:

grant you a 4th-level spell slot. If you have a spell repertoire, you can select a second spell from your repertoire as a signature spell. At 14th level, they grant you a 5th-level spell slot, and at 16th level, they grant you a 6th-level spell slot.

And for Master:

grant you a 7th-level spell slot. If you have a spell repertoire, you can select a third spell from your repertoire as a signature spell. At 20th level, they grant you an 8th-level spell slot.

None of them mention anything about swapping spell selection, so you cannot swap.

The Learn a Spell action refers to a situation where you "add or swap spells". You can apply it whenever you get to add a spell to your spell repertoire. As you are not swapping spells, you cannot apply it to swapping spells.

Retraining the feat

Everyone however can retrain feats (kudos to András), if the DM provides for a teacher. The rule for retraining feats states

You can spend a week of downtime retraining to swap out one of your feats. Remove the old feat and replace it with another of the same type. For example, you could swap a skill feat for another skill feat, but not for a wizard feat.

As you select your spells each time you gain a spell slot, and you would gain the spell slots anew, you could then select new spells for them. As you have to select another feat, you may not be able to swap one of them for itself by retraining, but you could spend two weeks to first swap it for another Archetype feat and then retrain the spellcasting feat.

If you have access to the higher level expert and master feats, and wanted to change a basic spell, this would be even more cumbersome, as you would need to retrain away and back in the whole chain (they are dependent on each other), but it would technically be possible.

Answer 3

They can replace spells by retraining the feat

Basic Sorcerer Spellcasting adds one spell to your repertoire for each new slot:

Each time you gain a spell slot of a new level from the sorcerer archetype, add a spell of the appropriate spell level to your repertoire

The feat adds the spells, so when you retrain to another feat, they are gone. If you take the feat again, you get to pick new spell(s).

How long does it take to retrain the feat?

The simple answer is 14 days, once 7 to replace the feat with something else, and another 7 to learn it again. This might be the literal reading of this section of Retraining:

You can change a class feature that required a choice, making a different choice instead

However, as mentioned several times the developers used natural language, not considering someone would want to replace the feat with itself.
But we have precedent for that in Magic Warrior Dedication:

To regain your abilities, you must discard your mask and obtain a new one of a different animal, then retrain Magic Warrior Dedication into a new Magic Warrior Dedication with your new mask

So you can retrain Basic X Spellcasting for itself, to change what it gives you in 7 days. Depending on your level, this can mean up to 3 spells.