Advice Needed: Early Release Request from Current Employer Due to New Offer

I’m currently employed at TCS and joined a new project just about a month ago. Recently, I received an offer from another organization that is asking me to join at the earliest. As a result, I’ve submitted my resignation.

However, after discussing with my current manager, he is insisting that I serve the full notice period, citing that my role is billable and thus cannot be released early.

Given the urgency from the new company and my personal financial constraints — where I'm genuinely struggling to make ends meet — I was planning to write a formal email to HR and the HR Business Partner (with my manager in CC) requesting an early release.

What is the best way to frame my early release request to improve my chances?

Apart from citing financial hardship, what other valid reasons or approaches can I include to strengthen my case?

Would escalating this to a higher HR authority or using internal HR portals be advisable in this scenario?

Have others been able to get early release in similar billable situations?

Any insights or advice would be greatly appreciated.