Wow, that is a handful to deal with. :(
After our toilet overflowed last summer, I accidentally found a slow drip behind the shower while looking for water to clean up. The drip was due to an improper PEX crimp & connection by the factory. Luckily no damage that I could see. Doesn't take much sloppy work by a worker to end up causing extensive damage. The thought of water leaks scares the heck out of me.
That tall piece of paneling should peel away easily enough if careful. You need to get it started somewhere like at the back next to the HWT where you can get a chisel in behind it and slowly work along the edges. Just don't use too much force because the luan plywood can snap. Once that's off, you'll be able to see what's in there holding things together.
If you absolutely have to cut any luan plywood, it's easy to cut with a utility knife with a utility blade. If you happen to damage any paneling, sometimes you can steal some from places like under-bed storage for ex. where nobody will notice.
Pull the drawer out and remove the drawer slides which will be held in place at the rear with a couple of screws and then you may get access to wood screws attaching 1x1 to floor and walls.
That's typical TT construction, kinda like a meccano set that is easy to disassemble. Just a minimal amount of staples and screws holding it all together. Definitely not like cabinets in a house where there are dado joints, biscuit joinery, 1/2-3/4" plywood, etc. When the time comes, installation is simply the reverse of removal... :)
You might want to check all the PEX connections in the vicinity, including the "swivel" fittings behind the shower faucet and at toilet.
Good luck and post photos of the progress if you can.
