Should I contact my client about a mistake at our meet and greet reagarding the drop off date the booked dates?

Do you mean that the booking request was for Friday-Monday (or what have you), but at the M&G you and the owner decided that a Thursday drop off would be easier, so you're wondering how to handle the 'extra' overnight?

If you haven't already accepted the request (meaning it's still in your pending messages), you can edit the dates to include that extra night. If you have already accepted the booking (it's in your upcoming messages), I would send an Extend the Stay request for the extra night. In the message you can explain that per your conversation at the M&G you're adding the additional overnight.

Regardless of how you handle it, I would make sure that the extra overnight is officially booked through Rover. If the dog is injured or something happens prior to the Rover booked arrival date, there might be issues getting support from Rover or coverage with their insurance.

It was likely an oversight by the client when booking. If the early drop off was more for your benefit, you can consider reducing your fee. Just explain to the owner that all of the dates you will have the dog need to be booked through Rover for everyone's protection!