No, there is no reason to ask them to book through Rover. Rover did nothing to generate this job lead for you, so why should you give up 20% of your earnings to them?
I sit dogs from Rover, but more of my clients come from my own neighborhood through word of mouth. Over time, the dogs have met each other and I am quite confident in their ability to get along. Generally I only boarded one dog at a time, but because these dogs are all regulars, I usually have a couple extra during peak holiday times.
You should always adjust your availability on Rover to reflect any other bookings you may have outside of the Rover platform. When my regular off-Rover clients requested the holidays, I marked my calendar as having no spaces for Rover dogs.
Oh yes, whenever a new dog comes into the mix, I have asked the newbie to come over when one of my regulars is here just to see how they'll get along.