Jamie, you could have a graduated rate. The longer a dog stays, the lower the daily rate. (Which translates into short stays costing more, which might discourage people from booking short stays -- or reward you enough that it's worth your while.).
I list graduated rates in my profile. I use a spreadsheet[1] to figure it out quickly. When I receive a stay request, I click the "edit price" and make a "one-time adjustment."
I wish Rover handled this better. In your case your "Day One" might be so high that it would make you look unattractive in the list of boarders. People have to read your profile to see how their daily rate might be half that amount (with a longer stay).
You can also archive requests to keep your availability open until someone actually commits to a stay. (I have read others say this is how they deal with similar issues.). I would communicate to the person you're available on a first-come, first-served basis and more than happy to meet/greet and book them. "In the meantime, I am archiving your request until it is booked." (I think when you archive it, it looks to them like it's being declined. So, you'd want to communicate something, I believe. You can still communicate with them in the archived folder, and book it from there.
[1] https://docs.google.com/spreadsheets/...