Hello,  It sounds like the piano action need attention.  If the middle
section is way louder than the bass section (from the test roll at low
pressure), then it is probably because the midsection centre pins,
bushings and felt, etc., are worn so that there is less friction there.

I am not 100% sure what "bobbling hammers" means but I assume it
means that the hammer strikes the string many times instead of once.
The cause of this is that the action does not release properly and the
hammer bounces on the jack(?).  If it does not do this when manually
pressing the key with similar force, then the pneumatic is not lifting
high enough or the whole action is adjusted for too much manual key depth.

You could back off the release a little or you could remove the free
play between the pneumatic fingers and the action, and/or you could
adjust the end stop of the player action to allow a little more lift.
The lift should be the same for manual and pneumatic operation and the
hammer should come to rest on the same place for manual play and for
pneumatic play with similar force.

Let us know how it went. :-)

Regards,
Bernt Damm
Sydney