This is a forum. You want legal advice, go find a lawyer lol...

Your employment contract between you and the company is what binds your relationship. Unless you agreed separately in writing with the company to amend your existing contracts, than 1 month notice (as stated in your contract) will suffice.

If your current manager is being difficult, you should deal direct with HR.

If you intend to exercise a buyout option, request your current HR to prepare calculations and what is the amount due. Once it is firmed up than you liaise with your prospective employer to issue a cheque for the amount. Before you handover the cheque, prepare cover letter and have your HR acknowledge the same.

Other strategies:-

(1) If they still hold you back, ask them to provide their reason in writing.

(2) Depending on the reason provided, you may either re-consider your termination notice or proceed to file a complaint with industrial court.