How to get an early release

Hey folks,

I'm currently working for a US client and I am 100% billable to them if that information matters. According to the offer letter, the notice period is of 90 days but there's a buyback option where one can choose to pay and not serve the notice period. But from what I've heard, it's up to Capgemini to accept that buyback offer or not.

So I've recently received an offer from a Singapore based startup and I told them that I can join in 30 days cuz if I hadn't, they wouldn't have even interviewed me in the first place. They're okay with buying back my 1 month and let me serve 30 days but I'm concerned what if Capgemini refuses to let me leave.

What are my options here? Can I tell them that I'm leaving company due to some personal/health issues so I can get an early release. Do I need to involve a lawyer?

Thoughts please.

Thanks.