The best chance is to contact/join the MG Car Club as they have access to some of the build records (although mainly for the MG Rover era cars I think - 2000 onwards, so I am not sure what is available for MGF models built under Rover Group before that date).
Maybe the above has already answered your question, but if not, "Maninthe car has got it right:. I just joined the MG Car Club and Registered my car with the MGF Register, and I now have a (digital copy only) certificate which tells me exactly (to the second! when it rolled-off the production line, and all the details about trim etc.
You may not get the benefits of local club meets and events over in the Colonies (sorry USA, only joking), but the Safety Fast magazine is a good read and there may well be other F and TF owners in your part of the U.S. The club can put you in touch with them.
I was interested in build date etc. because I had supplied and installed some quality testing equipment to the gear lab at Longbridge about 2 yrs before that. Perhaps that explains why the GG1 gearbox has a good reputation for robustness!