Wow, I had an almost identical problem. You can see my thread here:
https://terrylove.com/forums/showthread.php?p=57025&posted=1#post57025
Luckily in my case the leak was at the front of the valve (but still in/behind the shower wall) and the greenboard just below it soaked up most of the water. There was some wood damage, but luckily the structure is still sturdy and only the braces between the studs to support the faucet need replacing. I may add more anyway though. There is only one slight soft spot in the ceiling below.
I also have a very similar valve as you describe it, but the picture is way different. I've just decided to sweat in a new valve rather than repair.
Don't know if this helps; feel free to comment.
Thanks,
T.
Oh yeah, I don't have an access panel at all either. I might be able to build one that a smaller person could squeeze through, but I don't think I'll do that until/unless necessary. They'd basically have to crawl under the sink and into the vanity and then through a hole in the side of the alcove under the sink; I already plan to just get a cabinet face and not have a full cabinet under there anyway. Even then access would be limited and awkward at best due to the fact that the plumbing wall of the shower is also the wall of a chase, so there are ducts, flues, and other plumbing tubes in the way. Might be worth it just to have a small hole that someone could at least look through and shine a light, if there is line-of-sight from under the vanity to the valve.
Terrence