First issue Registered notice to the college under CP Act demanding for such Transfer certificate, wait for 15 days, then file on line grievance in Ministry of education pgrms portal. Hope this may work out.
Also provide a certified copy with service proof of your notice to the new college, with copy of grievance and give undertaking that you are making efforts and will provide TC and request them to temporarily postpone the requirement and grant admission.
Migration certificate is mandatory in between Universities and TC is different. The TC denotes that there are no fee arrears. Providing TC is manatory.