I suggest framing it as an orientation. Make it about their pet's well-being. "We want to make sure Sparky is content and safe here. Part of making that happen is having an orientation we can book right on Rover as a 'Meet and Greet.'" and then list your time slots.

I hope this helps! The other answers are definitely great for reminding us how crucial the meet and greets are on our side. Make it about your customer and it may help change their tune. Otherwise, say that the orientation is a requirement, and that you cannot take on their dog if they cannot do it. Gently, of course!