We haven’t had the same problem as you but we have had problems getting our college to use the right wording in the covering letter to go with the ESOL qualification. It does seem strange that your College are refusing to give you a letter and unusual.
Can we suggest you put your request in writing to your college. Write a letter to them. Keep a copy so you have something to show the BIA Border and Immigration Agency. In the letter spell out the urgency of the situation and that it is the BIA who have asked you to get a letter from them. Deliver it in person. Politely insist that you have a written reply so that you something to show the BIA. Getting things in writing generally makes everyone treat the issue more seriously, and gives you evidence to prove you done what you can to meet the visa requirements.
Best of luck.