First, congratulations and well done! Now, my biggest piece of advice is be happy/at peace with your final decision no matter what. You'd be ecstatic if you only had one of these offers, so recognize that you really can't go wrong here!
1. Is BCG really that superior when it comes to career/exit opportunities compared to Bain?
In SE Asia yes 100%. More exit opps, better reputation, more diverse projects, etc.
2. Is Bain's culture and sustainability really that much better compared to BCGs?
So, this really comes down to the individual. Both are known for caring more about sustainability. Culture depends on the individual here - they're for 2 somewhat different types of people! That said, you say the following:
"They seemed so much more fun and enjoyable to be around during the interview process, and it made me feel like working in that close-knit culture with these awesome people would be the best fit for me. "
Your decision: Ultimately, it seems you have to decide between prestige/ops and culture fit here. Both have their pros/cons...