I've received a lot of "we'd like our dog to be the only dog staying with you" requests (never followed with "they're aggressive towards other dogs" though) and recently they're preventing me from taking on other clients because of overlaps (I had to turn down a week stay because it overlapped for 8 hours with one that was already booked even though I would be home supervising).
If the owner is requesting a solo only stay when you normally allow multiple dogs, you should have the option of charging an additional fee for accommodating their special request. I would list on your profile that you allow multiple dogs to stay at one time, however if a pet parent would like their dog to be your only guest, it is an additional price you set and mention on your profile. You can discuss the special price with the pet parent via Rover messages and adjust your price depending on your agreement. Prices can be customized after a pet parent sends you a request and before they send their confirmation.
I only take one dog, or multiple dogs from one family, and I've actually had clients tell me they prefer their dog, or family, to be the only ones I'm caring for.
If you think you're losing business because of the solo-only stays, don't take them. Just say that you don't think they'd be a good fit for your business since you do offer boarding for multiples.
If a client brings up whether their dog will be the only dog with me (besides my own), I tell them that, yes, overnight I only take one dog at a time. However, there are times when a new client arrives a few hours before they are scheduled to pick their dog up. In my case, because I only sit new clients on weekends and holidays (when I'm home almost all the time), that would also mean the "overlapping" client is a returning client that I have determined is fine with my weekday work schedule, so I know the "overlapping" dog pretty well. I've never had a client be uneasy about that. If they said that they would prefer their dog absolutely be the only other dog for their whole stay, I'd probably charge more since they could cost me a week (or more!) of business from a returning client.
If you usually sit multiple dogs at the same time, and they want a solo stay, I think an upcharge is well-justified.
It would be entirely fair of you to give a custom rate, essentially allowing the owners to book all of your slots to ensure their dog is the only one in your care. What you charge would be dependent on how many open slots you have. If it were me, I'd consider charging something like a second dog rate (assuming you discount for multiple dogs from the same household) for each day for each extra slot they need to reserve; I feel this is a good compromise between giving them a fair rate for the work you are doing and making sure you don't lose potential income for being booked when you otherwise could have hosted additional dogs.
I've never thought about this. Definitely something for me to to think about! I only ever take one two dogs at one time (or three if two dogs are from the same family and I get one extra request; I have a stay like this starting in a couple days). I always let owners know that I take up to two at a time, except for that rare special exception and that I don't book the second overlapping stay unless I know all the dogs are friendly. I definitely think it's fair to charge a custom rate if they ask for a solo-only stay and you usually take on multiple dogs, though!
I would absolutely charge more. It's the same reason as with cancellations - by not being allowed to take on other dogs at that time, you're potentially losing income. If you want to offer this, I would put in your profile that you allow it for an extra fee. Then if someone brings it up, quote that policy ("I do consider only taking one dog at a time, but because during your stay I won't be able to watch other dogs, I charge an additional $/night, for a total of $ per night." etc)
Just have a firm policy, and adjust the cost of the stay manually before you book. If you have already booked and they bring it up, I would say again that you charge an extra fee for that, "as stated in my profile" of X amount..."
During past meet & greets, I've asked the pet parents how their dog gets along with others. This sometimes naturally leads the discussion that way. In the rare times when a pet parent has told me they prefer their dog to be the one and only guest dog, I've replied that if they want that, I can provide them with a custom rate, because I would not be providing care for other customers who may have overlapping needs. I've also shared that most of the time, I do not host not multiple dogs and let them know what is currently on my calendar (as in if I'm not expecting any dogs that time period or dates of returning well mannered guest and brief info. such as breed/age). I also reassure them that I do only host small dogs with no temperament issues, the same level of medical preventative care, and I'm always with them. I find it's easiest to discuss this in person at the meet & greet, after first ensuring the match between our dogs.
At a quick glance, it looks like your rate is on the lower end of the competitive market for your area, so it's quite possible that clients would be willing to pay more for a custom rate for one&only pet guest.
For the scenario you mentioned, if you knew another local Rover sitter who could accommodate solo guest, another option might have been to somehow work out sharing the stay - booking beginning date(s) with other sitter and then remainder with you (with the pet-parent agreement).