First of all, for house sitting or boarding, you can't look at the hourly rate. Way too depressing! But it is not like you are entertaining the dog for every minute.
Rover offers house sitting. Nothing is called an "overnight" service. Like boarding, each "night" is good for up to 24 hours of care. However, many house sitters also care for other dogs concurrently (drop-ins and walking) and will come and go from the client's home but they will spend the majority of their time (especially in the evening) at the home.
House sitting should be considered a "premium" service because you are not only watching a pet but also the home and may be asked to do little tasks like taking in the mail, watering plants, etc. A client should be paying much more for you to uproot yourself and move into their home.
In this particular situation, have you explained to the client that you have other dogs to walk during the day? Is the dog's medication the reason they want you to stay 24/7 without leaving? In any event, If the client's wishes are in conflict with your business model, then the client should be apprised. Tell them that your posted rate does not give them exclusivity. You will not be able to walk other dogs. That loss of income means this client should be paying more. I would easily double the nightly charge, but that should be based on you local area's norms,
I would definitely consider it. I think that is quite a lot to ask someone to not ever leave for that period of time.
I am a client that needs something similar. My dog would need them most of the day, and all night. I am simply having a hard time understanding how someone can only be asking $35 for this. How much should I offer?