The rhythm circuit only works on the neck pickup in Jags and Jazzes.
Get the switch fixed. it`s not that big a deal. Replacement switches are cheap. I broke mine a couple of years ago, and took it in to a shop for repair- It took only a day or two, and cost me only about $35.00 parts and labor.
Keep Jaguar pickups on your Jaguar. If you want more gain, try Seymore Duncan Quarter-Pounders.