Plumbing – Is it wise (or even code-compliant) to cut an access panel for the bathtub drain

It looks like you might be able to do this pretty easily.

To remove the stopper, simply unscrew the top of the handle of the drain stopper, then unscrew the underlying screw in the shaft to pull out the drain cover completely. Then, you'll have the cradle-thing that holds the stopper. You might be able to leave that in place (I grew up in a house that had the stoppers taken out but the "cradles" left in place, and two women with long hair would share the bath. We took turns cleaning out the drain) and place the mesh catcher over it. See this video for a demo (It should be region-free.)

You'll probably want to put down putty or some sort of long screw from the drain to the "cradle" in order to keep it in place.

Most access panels have a lip that covers a good half inch around the perimeter of the opening, so you don't have to make the hole perfect. Pick up your desired panel from the store so you have a reference. It should include a door and a separate part for mounting on the wall:

Start by cutting your hole slightly smaller with a drywall keyhole saw.

Check for any obstructions with this hole and adjust the position accordingly to make your final hole. Use the panel on the wall as a template to mark out the position of the final hole, and then cut that out with either the same saw, a sharp utility knife, and/or a rasp: