Taiwan however still wishes to maintain its own government and control of its own destiny,
That's a false premise.
Co-operation and co-ordination with the mainland does not equal capitulation.
Having a common, co-ordinated policy with the mainland wrt the SCS makes perfect sense for Taiwan, especially since the two have exactly the same claims, and doing so gives Taipei more control, not less.
The only ones who would oppose such a move are die hard seperistis who would rather see Taiwan razed to the ground like Ukriane rather than have it reunite with the mainland (most of these people would flee to the west in the event of war in any case, which may go some way to explaining their irrationally extremist stance as they won't be the ones to suffer the consequences, but I digress), those countries with competing claims and the US, who is hell bent on isolating and daemonising China.
Alienating China to curry favour with the US might be the default setting for past Taiwan governments, but in this instance, the logical course of action would be to side with Beijing over Washington.
If Vietnam invaded Taiping, would the US 7th fleet shed blood to throw them out? Not a chance.
Unless Taiwan has the means and the cahoonas to go take it back by themselves, the only alternative is to ask the PLA to step in.
Now, bear in mind that in the event of a Vietnamese invasion of Taiping, the PLA would almost certainly move in to retake the island if it looks like Taiwan is unable or unwilling to do so, Taiwan if it sides with the US, its choices would boil down to either loose Taiping to the Vietnamese or the mainland or have to fight a high risk and high cost war far from home by itself with minimal to no air support against Vietnam to take it back, in which case there is a significant chance it will loose.
That is if Beijing gives Taiwan the option of making a choice. It would not be outside the realm of possibility for the PLA to deploy before Taipei works out what it wants to do and take control of the island from the Vietnamese. If that happens, Beijing will hold all the cards and it will be a massive black eye for the encumbant Taiwan government even if the mainland hands the island back with a big pink bow on top.
The best way to ensure Taiwan remains in control of Taiping island would be to forge common ground with the mainland and either explicitly or strongly imply that all the islands controlled by Taiwan in the SCS are also under the aegis of China, and that any attempt to take any of those islands by force will necessity a PLAN military response.