I finally built a folding counter in the closet/laundry room. It's going to be a problem if I ever need to get to the valves in a hurry since the whole thing is nailed down, but I got tired of waiting and taxing my poor brain for a removable solution. I set a piece of 1/2" smooth-faced plywood set on top of a ledger board all along the edges. There is a 2x4 leg in the middle for additional support (to the right of the cabinet). If I had to do it over again, I would install the ledger boards about 3" higher. It would completely cover the valve box and would give me more room to slide out the appliances whenever necessary. I'd also work a little bit harder to figure out some way to get to the valves.
Obviously I still need to work on the face frame and the cabinet, but I like how it looks so far. I used Frog tape for the first time and love it. I've never been impressed with blue painter's tape and have been saving my roll of Frog tape for a project that needed really crisp lines. There are a couple of touch-ups I need to do, but those were all my fault - I accidentally dropped the tape in the still-wet paint as I was removing it.