What to charge for transporting about 20 miles?

Would I ask for the same under similar circumstances? No. Simply because I would not accept this request, here's why:

-I don't do pick-ups and drop-offs -I don't think the Rover guarantee would cover the transportation -The distance would be too far, would have to charge too much to make it worth my while

Also, as Walt said, you are charging far too little for daycare. Generally that price attracts only the type of client it would best not to have. The fact they're trying to negotiate price with you is a big indicator of that.

Know your worth. Set your prices accordingly. Never allow a potential client to try to bargain with you. If they don't like what you charge, let 'em move on.

The first thing I would do is raise my daycare rate sorry but $16 is just too low. Most of your local sitters are charging over $20, most around $25. Charging $30 for transport seems high to me given your base rate is so low, but as I recommend you need to increase it. What you keep after Rover fees has no effect on the client so you need to decide if you want the booking or not, as they have pushed back on the $30 charge. $36 for the day or nothing?

Yes, I agree $30 round trip is fair. In my area, I have charged that for pick ups/drop offs to a nearby sitter/friend who could not do that and was offering day care (which I was not offering).

Your day care rate is too low. It's attracting a person that is not showing they do not value your time. If you want this gig, you may try pointing out to the client that you're aware most sitters charge about $10 more for day care (so you're already giving them a great discounted deal), and your transportation rates are firm.

I wouldn't offer to pick up, watch for 9 hours, and drop off for a New client whose dog I did not already have a great relationship with for $36 ($21 after fees).