Can I accept jobs outside of Rover that began on the platform?

You will be violating Rover's Terms of Service if you go ahead with this client and not book it through Rover. If the client cannot afford a sitter's posted prices, it really is unethical for them to suggest this approach. You would face account suspension from Rover if they found out, but you are also not protected in case this person doesn't pay you or claims damages, etc. Don't feel sorry for the person either. It isn't your concern whether they can or cannot afford a sitter. That's their responsibility when they decided to have so many dogs.

dont do it!! Rover is amazing and so so helpful

I'd also like to add that Rover can support you when things get awkward. I also had a moment when I thought, well, maybe I should do this under the table. But an incident with a pushy client wanting me to keep her dog longer when I couldn't made my decision because Rover support stepped in to help handle the situation.