Does anyone know If I use the "rebook" option for regular customers, will my clients still have to pay a Rover service fee?

Yes, they will still have to pay the Rover service fee. There's no way of lowering costs for your customers, except at your own expense.

I don't know why you want to lower costs for them. Have they asked or is this just something you want to do? If you want to institute some kind of frequent customer reward program.

Maybe you could make up little gift bags for outstanding customers. Again, this is out your pocket, not Rover's.