How can I reseal a shower drain on a pre-formed shower base?
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.