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.
I disagree with BMitch as far as if the P-trap will come apart. (Possibly the first time we've disagreed?) The metal nut could not be where it is if the joint did not separate. The rest of his answer I agree is the easiest fix. However, if possible, physically cleaning the trap is preferable to a chemical approach. I'm assuming there is a slip fitting just out of the picture, just as there is in non-disposal side. By detaching the metal ring in the first photo and the unseen slip joint, the trap should be able to be removed.
Even without a slip joint, the pipe could be detached right at the disposal itself. This could actually be preferable as it would allow the P-trap to swivel, which may help in loosening the nut. Are you sure you are trying to turn the nut the correct direction? I'm not saying you don't know how to loosen fasteners. The direction depends on which piece the nut is threaded onto, which is not always clear in plumbing.
The curved shoulder of the nut tells me it is threaded onto the U-bend, so the nut is turned anti-clockwise when looking down at it. It's hard to tell by photos, but you appear to have decent access to the nut from the left side as it is below the pipe coming in from the other side. Even though the opposite trap may be in the way for fully turning a wrench, you seem to have enough room to place the wrench handle near the trap and pull right.
As for counter force, if pulling on the pipe itself isn't working, try a large screwdriver wedged in the inside of the U-bend. If you have the right wrench, the ring nut shouldn't offer a lot of resistance.
Best Answer
Use a large pliers or pipe wrench if you can't turn it by hand. Lefty-loosey, righty-tighty (from the perspective of the part without the male threads--top down for #1 and bottom up for #2).