Sounds like the expansion valve to me. Short period of cooling then it cuts out and warm air only.
Had the same problem with my previous Supra.
It's not a difficult fix as the expansion valve can be accessed by removing the Air Condensor from behind the dash. The expansion valve is at the rear of the Air Condensor. No need to remove the dash, can be done by removing the lower glove box.
If having this done have the receiver drier replaced too. Parts are relatively cheap from factor places. If you search under receiver drier you will find some threads about this.
I had all the work on mine done by a specialist which cost approx £400. You could get the system vacuumed by Kwik Fit or similar, have a pressure test at the same time, then fit the parts yourself and return for a regass. That would save some money for sure.