Will they add an antifungal as a hail Mary? I normally wouldn't advise that since it's a pretty toxic class of meds, but if Sulfatrim is first on the list and it's not working, then most likely something else is going on too. You're between a rock and a hard place because he can't wait for the culture and can't tolerate further tests. If he were my bird, I'd want him on a combo of antibiotics that covered the broadest spectrum possible, plus an antifungal. I hope it's not the case, but you may just be out of time.
Btw, vets legally can't prescribe to a patient they've never seen, so I'd put my time and energy into either getting him hospitalized somewhere or working with your established clinic via the techs and whoever's doing coverage.