I don't think stick-on tiles will work on top of rat fur, and I think cutting away the rat fur (and Reflectix) is a bad solution. The shell lining serves to deaden sound and control condensation, as well as insulate.
I agree about using a lightweight faux tile panel on top of the rat fur instead. This isn't a house. The optional Scamp
factory backsplash is a piece of thin fiberglass over the rat fur riveted in place with the cabinets.
To install it you could do what Scamp does. Drill out the rivets on the top edge of the base cabinets and the bottom edge of the upper cabinet, slip the backsplash in between, and put new rivets back in (might need to be slightly longer. not sure.
But personally I would take the easier approach suggested by Gordon of just slipping the backsplash panel between the cabinet edge and rat fur without removing the rivets. The lining has enough give to allow you to do that. If the vertical edges bulge away from the rat for a little, use a few self-adhesive Velco hook pieces to snug it up as Kenneth suggests. You may need a couple near the window, too, to keep it looking neat.
If you're really determined to have the tiles, install a thin backing over the rat fur in the same way (probably riveted due to the weight) and stick the tiles to that.
I also agree with Gordon about only light
cooking inside. Fry your bacon and fish outside, or the smells will linger on the rest of the rat fur, the cushions, your bedding...