Your approach is correct. Look for a switch than can handle the 10 amp load and voltage. Lots of switches that match "20A 125V DPDT" on the river site. I also find these at my local hardware store.
If you tie the transformer's grounds together AND verify the transformers are in phase (always a good practice), then a SPDT switch for the "hot" wire will work. A 3-way light switch suffices and the parts to build a box for it are off the shelf.