My interpretation of the rules as written would be yes, you can swap spells even if the gained spell does not appear in your tradition's spell list, however, you may not be able to cast them.

My conclusion was based on the following:

Definition of repertoire

The CRB defines repertoire for both Sorcerers and Bards as follows:

The collection of spells you can cast is called your spell repertoire.

This is quite broad. The text explicitly states that any spells gained from leveling must be taken from your tradition's spell list, but does not impose any conditions on gaining from other sources. In fact, it explicitly states that your spell slots and repertoire are not linked:

Though you gain them at the same rate, your spell slots and the spells in your spell repertoire are separate. If a feat or other ability adds a spell to your spell repertoire, it wouldn't give you another spell slot, and vice versa.

Learn a spell

The description of the Learn a spell action is very explicit:

Requirements You have a spellcasting class feature, and the spell you want to learn is on your magical tradition’s spell list.

You can gain access to a new spell of your tradition from someone who knows that spell or from magical writing like a spellbook or scroll. If you can cast spells of multiple traditions, you can Learn a Spell of any of those traditions, but you must use the corresponding skill to do so.

Given that Arcane Weaving does not reference this action, it explicitly does not apply.

Arcane Weaving

Finally, since the Spell repertoire class feature does not impose any restrictions and we're not using Learn a spell (which does), we fall back to the text of the ritual itself (emphasis mine):

You and the secondary casters weave together spells (and, if anadis, silk) in a complex ritual that combines magical learning with art, allowing all the participants to share spells they know with one another.

Success You or any secondary caster can swap any spell in your spell repertoire for a spell in the spell repertoire of any other participant. This spell can't be a higher level than the ritual's level.

The rule for this ritual specifically calls out any spell with the only restriction being that it can't be higher than the ritual's level. Additionally, the flavor text for the ritual seems to imply that this is a magical transfer of knowledge, which would presumably include any tradition specific knowledge required to cast the spell.

Casting

Now that we've determined that we can learn the spell, can the character cast it? My guess is it depends.

Looking at the Occult spellcasting entry for Bard, we see (emphasis mine):

You can cast occult spells using the Cast a Spell activity [...]

As for the sorcerer, the Sorcerer spellcasting entry states:

You can cast spells using the Cast a Spell activity [...]

This seems to me like it may be an oversight in how the class features are written. Personally, I'd allow it in my game, but definitely check with your GM first for their interpretation.