Looking at that trigger stick.... why couldn't you route out ~4mm of the wood panel and then sit the plate in that? Simply use normal headed machine screws and then the heads won't be above the surface of the wood panel. Then just two pieces of 3mm plexi to sit in the gap you have left. You'd need to cut little bits out where the machine screw heads sit, but the rest of it will now be flush. You need two pieces of course because you need to put them together round the base of the stick (with a hole cut out when they meet together)
You may need to adjust the measurements I've said here. For example I'm guessing that the mounting plate is 1mm thick. And I'm assuming that the stick will still function correctly if the plate is recessed by 4mm. If the plexi isn't perfectly flush (I'm a bit anal when it comes to this sort of thing

) then I just place bits of card under the plexi until it is.
I've done this method before with joysticks to get the ball-top at the perfect height for my personal preference. Then I just fill the recess above the plate with a piece of plexi to make the whole panel flush. Once artwork is put over the top it's no different to any other panel.
Just don't permanently affix the artwork so that you can get to the screws if you need to do any maintenance later. I use small bits of double sided tape to hold the art in place. It's firm enough but easily removable later if needed.
EDIT: I just had an afterthought... I take it the handle of that trigger stick is easily removable yes? (I've never used one before to be honest). Because if not then applying artwork is gonna be a ---smurfette---! And if it is removable then you can fit one piece of plexi in the recess with a hole cut out... no need to make it two pieces.