I'm not a Plumber but do my own plumbing at home and can easily sweat copper.
What about simply 4 joints per line to sweat instead of 2 by removing more of each line than just the existing couplings?
Picture the valve positioned on either side of the cuts with a short stub of piping already sweated on while using a slip coupling to connect the stub to the remaining piping.
If space is extremely limited the stub could be slightly longer than a slip coupling, so that the coupling could be slipped on the stubbed out valve while being fitted in place.
As for sweating near wood, buy a Oatey 9 in. x 12 in. Hands-Free Heat Shield from Home Depot or the equivalent from Lows.
You could also soak the wood with water from a spray bottle if you're proficient at sweating a joint.
If you didn't want to sweat, this plan could be replicated with Sharkbite fittings, only if you have the increased length needed to accommodate their longer slip coupling.
I only have 2 Sharkbite fittings in my home and they are at the top of the flexible supply lines for my water heater.
I would not install sharkbite fittings in a wall, but if that's an unfinished basement space that you're in frequently enough to notice any future issues... then you might feel comfortable using them there.
In my opinion from the pictures shown, there will be plenty of play to use either regular instead of slip couplings as long as the wall where those lines exit to your laundry area weren't foam sprayed or sealed in some other manner.
With the idea that every well thought out project requires two trips to the home center anyway, just buy regular and slip couplings and take back what you don't use.