First off, I've never used Motioneyeos before, so I apologise for the terrible method in which I go about solving this. Secondly, I'm looking only at the wiki page for Motioneyeos for my ideas, so if there's any mis-match in what's really there then have a go tweaking around everything.
Anyway, it looks like the action buttons just run bash straight up, yes? SInce the PanTiltHat is grounded mostly in Python (and since it's my favourite language), it should be simple enough to set up four Python files that include code to move the hat in it's respective axis for a set amount of time, which are then ran by the bash code from the action buttons. For instance:
sudo python2 LEFT.py
Again, sorry if this is a complete waste of time, but I hope it works anyway! :)