Strapping down a copper pipe with little access

I installed a new tub faucet because the old one had a terrible diverter.

Because the pipe moves in and out (about 1.5inches), it’s impossible to get the new faucet to fit tight against the tub. I also can’t caulk any gap because the movement will break the seal.

I have a plan to fix it and I want to make sure this is the best plan.

I’ve opened up the wall to strap down the pipe.

This is what it looks like inside:

The pipes run along the inside of the studs so I cannot screw in a strap.

So here’s my plan and I’d love feedback. I plan to add a 3/4” thick piece of scrap plywood behind the pipe. 3/4” is the distance between the tub and pipes.

Then, I’ll strap the pipe to this scrap plywood with this:

That will keep the pipe from pulling towards the inside of the tub.

Then, I’ll toenail a 2×4 between the studs. Finally, I’ll screw through the 2×4 into the 3/4” plywood.

Is this over-complicating things?