The reason you can't fly using GPS derived altitudes has nothing to do with correcting for pressure, as alluded to the GPS altitude is not measured using barometric pressure. However due to geometry the altitude solutions for civilian grade GPS receivers are generally not as good as horizontal positions.
There is some confusion over this because
TSO C129a requires the pilot to "manually insert the barometric pressure" unless "the automatic altitude input utilizes barometric corrected altitude data".
For the OP, the maximum reportable altitude depends on the chip used.