I would use a phosphoric acid based toilet bowl cleaner instead.
Phosphoric acid is the active ingredient in many bathroom cleaners because it cuts through soap scum like a hot knife through butter, but doesn't attack chrome.
My own experience is that phosphoric acid at about a 15 percent concentration is a lot less aggressive than CLR in terms of acidity.
The gelled aspect of toilet bowl cleaners would allow them to stick to the vertical surfaces of your shower better so that you only need to squirt the phosphoric acid toilet bowl cleaner on, and then use a scrub brush to spread it all over the walls. Then, use a rubber squeegee to relocated it to the grout lines and give it time to work. Then turn the shower on and rinse it off.
DuPont makes a number of stone and grout sealers under the name "Stone Tech". I would trust that DuPont has the knowledge in chemistry to make a long lasting stone or grout sealer.
DuPont? StoneTech® Professional Protect