Heavy Duty 16 Gauge Stud Shoe, MUZIK Stud Shoes 2x4 & Stud Shoes for 2X Studs - 1.5"x6.25" Galvanized Steel, Fits 2-3/8" Pipes, IBC/IRC Code Compliant Pipe Support

I did the same for my laundry room/garage entrance remodel for the reason you have. I have a LG WKEX300HBA unit and very happy with it after overcoming machine smells for a while. You will be fine moving the box. If you worry about the making a big hole on wall stud, you can add a stud shoe in the link below. You may position the dryer vent higher to accommodate the upper dryer vent with a galvanized pipe. And make it recessed such that connection port is pointing up inside the recessed box (this will save a space behind the wash tower unit for vent hose). There are some nice locking or magnetic connector. I prefer this locking one. Lastly, You may want to move the washer outlet too if possible, although current position will probably work. dryer vent stud shoe