Sorry, have to disagree.
Pilot selected baro IS DOWNLINKED via Mode S where it is available. Barometric pressure setting is part of BDS 4,0 - the transponder comm-B register that also contains Selected Altitude. Unfortunately, European policy makers in their infinite wisdom decided not to mandate the baro setting as part of Enhanced Surveillance.
As such it is not displayable to ATC as it could not be counted as being reliable data.
A shame really, as it could be a useful piece of data to provide a controller tool to indicate a pilot operating on wrong QNH when transitioning from 1013mb.
RS