Have you contacted American Standard? When I was trying to remove, what I thought was a Moen part, I contacted Moen and they had me send a photo. Turns out it wasn't a Moen fixture, but they guided me toward the correct brand.

That's a pretty cool shower kit......sure you don't want to keep it? : )

I had to replace parts to Delta shower/tub kits in a 70's house and it wasn't easy. I was limited as to the styles that would work with the existing valve and even tho Delta said it would all work it didn't. I ended up having to use some of the old parts along with the new to make everything fit.
So, don't throw anything away when you start taking it apart since it may have parts you'll need that are no longer available.
Also, work with a plumbing supplier (not a big box store) in finding what you need.

This is not modern?

I've never seen anything like this from Am Std. IMO there is little chance of finding anything that would replace the trim; you'll probably have to remove the valve from the other side of the wall and find a cover plate large enough to cover the exposed holes.

If it's working, I wouldn't touch it.