I rebuilt or basically built a small TT and used 2x2 tile for the floor and backsplash. I used the flexible premixed grout and a colored silicone type grout for the joints. It was fine for 5 years that I used it and it sat through temp extremes. I would use the flexible grout/adhesive and a silicone in the joints not the traditional sanded non sanded grout. I cant see how a rigid cement type product could hold up. The joint silicone comes in many colors at box stores.
I would try sticking a tile and see how aggressively it holds. If it's questionable I'd then use something like goop to glue them on. Follow with a silicone grout. I stripped my carpet, primed the plywood and used peel and stick tiles. In the most difficult areas around the steps etc I did have a little trouble with some of them not sticking well so I used goop on them and had no more trouble.
If the back splash base material goes behind a cabinet into the cupboard area you may be able to test in an inconspicuous place before doing anything.
I think some of the silicone will work but you would want the type that's tough not easy to peel off. it may be labeled as adhesive silicone. Another thing that I have used is proflex and it's really strong.