Plumbing – Loop-de- loop waste pipe

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.

If you use flexible hose, and take the sharkbite fitting off the drain, the flexible hose will/should clamp right onto the 1/2 inch copper pipe. Or, you (or a plumber if you don't solder as I might expect for sharkbites) can solder a brass/bronze barbed fitting there, or I suppose you can build some sharkbite contraption that ends in a barbed fitting.

Hint, use enough hose onto the pipe to allow placing TWO hose clamps on the pipe, and put the screw-clamp part on opposite sides.

Alternatively, you could investigate what sort of threads you get if you unscrew the plastic barbed fitting, and look for a PEX fitting that matches that thread.