sure, this should work...
please note, the Push stuff i did was never 100% completed, as during its development Ableton launched the Push 2, with a trade-in offer, so I took that... so I didn't have a push 1 to work with (and the Push2 does not work due to being usb 2.0)
I hoped one day to pick up a cheap push 1 again for axoloti (and other projects) , but unfortunately (for me!) they actually have retained their value pretty well - so its not happened yet.
anyway, as far as i remember it should work, just things like the parameter display are not as good as they could be.
(and Im pretty sure, when the next version of the Axoloti software comes out, it will break this!)
Pyramid works nicely with it, you can even power the pyramid from the axoloti, if you use the usb port.
when i got the Pyramid initially, I did a few tests with this, as it seems like a great combo
not only, could you use the Axoloti for voices - but you could also use it to given you another midi din in and out.
(and as you have already noted, a USB host port)
unfortunately, I got distracted (*), and never got around to setting it up properly...
so if you create a patch, let me know... it would be an awesome partner to the pyramid.
(*) I started looking into writing a FM drum synth to use on Axoloti for the Pyramid, and that led to all sorts of other ideas, which... and I lost my focus (not an uncommon thing for me )