Have you considered mounting the stick and throttle to pieces of MDF that can then slide into a rest or holder of some sort?
Even just a groove to slide the whole thing back into, then a pin or lock/clamp to stop them moving around in it. I'd draw a picture and share it but I'm not signed up with any photohosting sites. I'm sure you get the idea
PS - Are you leaving yourself some time to actually PLAY the game, or are you only focused on building right now?
Yes, that's exactly the approach I've been contemplating! If you look at the original picture, you can see 3 points of attachement indicating my initial view on how this would work. My ideas have evolved as follows.
Option 1:
The shaded parts show where would pieces will be attached in the side frame to stabilise the removable throttle. This will come away in 2 sections with the frontmost piece including a platform to support the front of the throttle. The prongs would be hardwood batons (banister spindles) screwed and epoxy'd to the base of the throttle container.
This is the best looking option in my opinion but requires the most engineering but may be less stable than option 2.
Option 2:
As option 1 but uses a 3 pronged approach. The disadvantage of this is that it will require a greater degree of accuracy to have everything line up correctly and actually result in the unit being removable and not getting stuck.
A variant of options 1 and 2 would be to use 2 mount points rather than 3.
Option 3:
This uses a shaped panel to slot into a retainer. This is the easiest design to implement but will need to use MDF (double piece of 18mm). The shape has been designed so it remains tight with no movement fore and aft; only an upwards movement will be possible i.e. when removing it. It will be difficult to attach the lower shaped section to the base of the Throttle container (MDF doesn't like screws in the edges) and remain stable so may require further pieces of wood to keep it stable and prevent flexing.
Option 3 is my preferred approach at the moment . I will be able to make some cutouts in the lower panel for a better finish.
Just a view on my thinking so far ...
Yes, still finding time to play
Edit. It the removable approach doesn't work or isn't sufficiently stable, I'll simply hard mount.