Walking dog customer wants to set up a contract

I was walking a couple's dog through Wag when I offered to walk their dog at a cheaper rate outside of the Wag app through which we met. I've done this a couple times but this is the first time where the clients wanted to write a contract. The husband, a lawyer, would apparently write a contract that would "protect me as much as it would protect them" and so they asked for my ID. Should I agree to it or go back to walking their dog through Wag? I'm leaning towards the second option. The US is way too litigious for my tastes.