Strapping down a copper pipe with little access
No, I don't think you're over-complicating it. You don't want to just have a piece of 3/4 inch plywood floating around in there without securing it to the studs. There are a number of ways to do this type of job and your approach seems sound. Be sure to securely attach the 2x (screws or nails) and the 3/4 ply to the 2x. 
Also, I noticed what appears to be moisture around the lower run of copper where it goes through the wall. I'm assuming that is the feed for your tub faucet. I'm further assuming that the moisture is from prior unsuccessful attempts to seal around the tub faucet and isn't a leak. If that's the case make sure you allow that to dry out thoroughly before you seal up the wall. Might want to use a hair dryer on it but remember the moisture is permeating the entire width of the drywall - don't just get it surface dry. Even with a hair dryer I'd leave it open for a day or two. Good luck.