Ok, just tried the following: T-GT II wheel on T300 wheelbase (PS-compatible Ferrari Alcantara edition). No pedals connected for this test which I don't think will influence the result. None of the wheelbuttons seem to be recognized, most notably the PS-button won't work so the controller can not be used on the console at all.
Looks like whiskeypit was right however it's possible there's a firmware conflict in my case since the T300 has a pretty old one installed from long before the T-GT II's release.