LLigetfa
DIYer, not in the trades
It may be possible to "weaken" the existing blocking holding the valve rough-in by cutting saw kerfs part way through the blocking and then sister up more blocking behind it (if there is room) so that you can pull back the existing blocking. If there is not room to sister more blocking, you could drive a bunch of screws into the saw kerf to spread it and make the blocking bend back. BTDTThe other side of the wall is drywall. It's the master closet. I certainly wouldn't mind having a panel there for permanent access. Sounds like a clever idea.
This cutting of saw kerfs and driving in wedges is something I sometimes do on crooked studs that create a bulge that affects tiling or cabinet placement. I cannot take credit for it as I learned it from someone else.