Barometric aiding is certainly available on TSO'd GPS receivers, but it is not a requirement. Sufficient satellites in a sufficiently helpful distribution in the sky will also provide RAIM. RAIM is required for a GPS approach, but the rules don't state, and the pilot won't know, how the GPS receiver achieves the required integrity and accuracy for RAIM to be available.
I doubt ATC would know or care whether I'm using barometric aiding on any given approach. The whole point of RAIM is that it is Receiver Autonomous - that is, it requires no monitoring or intervention from outside the aircraft.
If an aircraft ceased squawking Mode C on the approach, ATC might tell it to go around for separation reasons - I wouldn't know - but not for GPS reasons I think.
Cheers, O8