With the signing of the referendum 3 years ago, will Bougainville ever be added to the profile flag selection menu?

Tokelau has been around for a bit, for example, and it's in the same "dependent territory" status.

I think your best bet to have it added would be to submit the feedback form in the support hub:

https://www.speedrun.com/knowledgebase/supporthub/give-feedback

Good luck!