(This is not legal advice. You're advised to seek formal advice if necessary).

Just offhand and purely based on what you've told us, I'd say that the recruiter's position is not strong.

The letter of commitment is just that, a letter of "commitment". They will obviously try to characterize it as a breach of contract but I really doubt it will be seen as a contract between you and BGC if this goes to court.

The real contract is between the Company A and BGC: the former is paying the latter a specified sum of money for successfully placing you with BGC; probably BGC's commission is 3 months and a % of your salary with company A.

There are many issues with making a commitment between you and BGC contingent upon the employment & salary between you and company A, which is a matter between you and company A alone. What if your negotiated salary with company A was $1 a month? What if company A was the one that wanted you to leave and gave you 1 month's notice?

My first advice is to wait out the 3 months. If that is not an option, go ahead with the resignation but threaten to go on social media if they threaten to enforce the "commitment" letter. Recruiters are highly sensitive to reputation.