Firstly, it takes a bizarrely small box to interfere with GPS signals
I envisage a transition period where GPS and baro run concurrently until the bugs have been ironed out. Local QNH would be set on the baro altimeters, periodically updated. There would be no transition altitude. If GPS is accurate enough for instrument approaches it's accurate enough for vertical separation, assuming equal accuracy in all three dimensions.
Secondly, Airliners (wrong forum I know!) fly efficiently at a certain pressure altitude. Changing levels just because the temp changes isn't going to be great.
Not sure this is relevant to vertical traffic separation, which is the rationale behind the Standard Pressure Datum setting.
In the UK we frequently have problems when low pressure weather systems cross the area. Crews mis-setting or neglecting to change altimeter settings is a headache for ATC, the problem compounded by the associated turbulence distracting crews at critical times.