If I understand the problem correctly, you have a plate installed on the top of the elevator already and have covered the elevator.

Keep in mind that I have not done this, but here is my suggestion of a possible solution that I would try.

If the plate on the top is not bothering you (it is flush with the covering), then you could make a thin ply plate the size of the control horn base, trace its outline onto the bottom where the horn mounts, cut away the covering, and retro-fit a plate on the bottom.

If you can hit it with something fuel proof (if needed) to color it, fine. If no color paint will work, cover the plate and leave the bottom wood bare to affix it to the elevator. Then mount your horn.

You will want to use the bottom horn to locate the position of the holes to be drilled, though, because you won't have any leeway on that side. Good luck, Dzl