Switching the low voltage should work exactly the same as switching the high voltage side except the current will be higher. For your application that shouldn't be a problem. However, you will have to check what power the relay board itself needs and make sure the relay switches both the lines (red and black on your diagram) or you could run into some serious safety issues. I suspect all that relay does is switch one of the three screws on the right to one of the others, you really need a module that switches two independent lines. (A double pole relay)
Brian.