Shower – Cover/block square Schluter Kerdi tile-in shower drain

Don't bother trying remove the drain. Don't try to renew the putty, you'll just have the same problem again.

Remove all the old putty you can. Roughen the surfaces that contacted the putty with coarse sandpaper. Clean the shower base and drain surfaces where the putty was a well as you can. Fill the resulting gap with self-leveling polyurethane sealant. Do so in a way to not trap air and to fully fill the void. Once that stuff cures, it'll never leak again!

If you do have to remove the drain in the future, simply cut thru the sealant with a sharp knife.

Shower tiles are not a place to cut corners. If they are porous then you will have a host of issues. Basically you will breed mold faster and the tile install won't last as long - mold gets in cracks and pops tiles out or at least loosens bonding. You need to get a non-porous stone, plastic, glass, or porcelain.

If going with 12x12 tiles you will need at least 1/16th of an inch given they are uniform - and perfectly cut 12x12 are not cheap. So you are looking at 1/8-3/16. Which means you need sanded grout.

For cheap I suggest you go on craigslist or whatever is equivalent locally (even try job leftovers at tile store). The bottom 3 feet of your install needs to be a good tile - you will need about 34 sqft. You can get decent porcelain for $2 a sqft and even cheaper if it is a job leftover or end of stock. You can still use your ceramic up top. And when switching tiles I usually put in a narrow trim of glass mosaic in between - will need 3-4 sheets.