FWIW, when I work with both H2 and Ardour, once I’m satisfied with the H2 “song”, I actually export it as MIDI file from H2 and import it in Ardour.
I find it simpler than going through the hassle of recording the MIDI.
Well, it does output MIDI ! and I can record it in Ardour if I want ! It’s just a bit of a pain to setup …
In H2, in the instrument panel, you have to make sure each used kit element has a channel number defined and the expected midi note.
MIDI channel is OFF by default, which most likely explains why you’re not receiving any data in Ardour