The floor covering mfgr meant the type plywood in your picture with exterior glue. Many floors are partical board and we know it's not exterior grade. The primary objective is a smooth surface that will not "telegraph" imprefections through the floor covering. Secoundary is having it resist warping. With the plys glued to prevent seperation the first ply can't seperate and warp under high humidity. Your contractor should have explained that to you instead of alowing you to fret over it.

I'd use the 11-ply douglas fir 3/4" plywood, and get rid of the 1x6. It's probably seriously degraded around the WC anyway, and the fir is tough as nails, extremely stable. You don't want to raise the floors so much that there is a step into the bathroom after tile.
You do not need to use a cement board, either; I use Ditra mat by Schluter.
Casey

A contractor that cant source ext grade ply?? Any lumber yard carries exterior grade ply and subfloor. You only need plywood that has exterior glue so the laminate don't separate. Drive past HD and head to your local Lumber yard and tell them you are wanting plywood sub flooring; fire your contractor; install the linoleum as per manufacturer's instructions. The APA marking is just showing which engineered lumber grading company the manufacturer belongs to.