I'd suggest you first need to know approximately when he'll pick her up. Then, you can include that in the booking. I also figure out the rate based on 24 hour increments. So, no matter what time the dog arrives, I can easily calculate how many hours the pick up time is beyond the arrival time, even with multiple days, as it can be counted fairly easily that way. I do offer a bit of a grace period (at least 2-3 hours generally) before I charge a partial or full day rate or additional overnight. Some things I take into consideration are the length of the stay and (with repeat customers) the frequency of booked stays.
For whatever it's worth, I would never agree a meet & greet that late at night (because I can't see the dog interact with mine outside well in the dark, introductory barking would not be appreciated by my neighbors, and safety reasons).
Some dogs need time to adjust to a new location, especially if they have a bit of separation anxiety. From experience, I've learned some dogs that arrive around dinnertime can't even unwind by bedtime, So I understand why you wouldn't want to take overnight at 9 PM, but can also understand why someone would rather do so vs. accept the dog at 5 AM, ...and I do know a sitter that would not count that night but start the next day just because evenings are easier for her schedule than mornings. So, you may have personal preference to consider in your decision too.