Here are some test runs as suggested by Dan.
The first pic is one of my early test panels done some days ago and shows what I want to avoid. On this panel I butted the foam at all four corners at a 45 degree angle. The arrow shows the shadow or telltale sign of this joint. When I put this panel together I knew the pieces of foam were not butted up tightly and I made no attempt to sand it or even it out. And this was the result. I think it is even more visible to the naked eye than to the camera.
Here are the tests I did today butting the foam pieces tightly using 3M Super Trim which did give me a bit less initial bite when the pieces are laid down and allowed me to form a pretty tight butt joint. I also then sanded the top surfaces paying special attention to the joint area. I did these tests with no muslin over the joint. The butt joint runs from top to bottom just about at the center of the vinyl in each shot.
I did two separate panels as pictured and the third shot is a close up of one of the panels. These were taken immediately after laying down the vinyl so I'll delay final judgement until the panels are completely set up and dry. But I can't detect any hint of the joint on either panel at this juncture.
Assuming I can create this tight of a joint between each section of foam on the actual panel, I should be in pretty good shape using your suggestions.