You are over thinking this. Get some 5/4" x 2" Pine, paint it, screw it to the wall and apply the mirror

Millworkman - so we don’t have to worry about the wood expanding and contracting? Also I was concerned about the surface size of the wood that screws into the wall.

Verbo - good idea about the medicine cabinet. I don’t know that we will find the size we need and we don’t want to see the back with the lights. It needs to be smaller than the mirror itself.

"so we don’t have to worry about the wood expanding and contracting?"

No, seal it well with a good primer/sealer and I would use a quality paint and it should not be an issue.

We bought this composite wood but started questioning if it would work. Thoughts on that? Is pine a better way to go?

That should work ok as well, I would mock it up carefully as I was thinking 5/4" thick which would net out to 1-1/16" in thickness, thinking the 3/4" (11/16" new might be too thin for good backlighting).

Gotcha. Maybe we double it up.