Not to be critical, but you're talking about a very difficult process. Making your "bowl" will be difficult. You might want to make the bowl by screwing real wood to the sides of the control panel just temporarily while you work with the epoxy. After it's dry, pry the wood off of the epoxy. When the wood sticks, just break it off and file it away. You'd have to try to polish the epoxy after getting all of the wood off. Then if you'll have an exposed edge of the plywood you can put the laminate on there.
You might also consider still going with Plexi on top. Use a small layer of the epoxy to suspend the coins, and then top it off with 1/8" plexi for a smooth top. You'll hide the imperfections of the epoxy top (unless that's what you're lookign for) and you probably won't be able to tell it's layered.
Next, for the mounting of the joystick take a look at the attached image. I would route the bottom of the control panel so that only 1/16" remains of the top layer of birch. That 1/16 is just to cover the original mounting plate which you would mount to a wood panel that gets screwed to the bottom of the top panel. I'm not explaining it real well, the image is better.
Another alternative is to use 1/2" plywood for the CP. Route out and recess your mounting plate on the CP like you normally would. Then use a birch laminate (I got a two foot by four foot roll at Home Depot for about $25) on top of everything, including the mounting plate. Your joystick mounting plate would be sandwiched between the laminite and the CP. Then top it off however you want. This would probably result in the best looking solution for the bottom. Again, these things are not exactly easy to pull off, and most likely end up in disappointing results. Trust me, I've got a basement full of disappointment

. Hope something I said helps, good luck.