Owner insists their dog sleep outside in crate, I agreed but now I'm a little uncomfortable. Should I bring him in at night and just not tell them?

Since it sounds like you're boarding him at your home, I believe you need to do what is in the best interests of the safety and well being of the dog. I'd advise you consult with Rover (call them -- many of them are sitters too) to get their opinion as they have a vested interest in the customer/dog and liability with insurance. If I was boarding a crate trained dog at my home, and the pet parents wanted the dog to sleep in the crate, the dog would sleep in the crate, which would be inside.

In my area, I absolutely would never leave a dog unattended outside, whether in a crate or not, because there is wildlife (raccoons, opossums, coyotes, owls, hawks) which would be scary and dangerous for canines. Even if I was house sitting, I'd suggest any crated dogs are kept indoors. When a pet parent asks me to do something I'm uncomfortable with, I tell them before accepting (such as "I understand you're comfortable walking the dogs unleashed. I am not. All dogs I walk must be leashed".)

Everyone's answers are great, and I definitely agree that you need to reach out to both Rover and the owners.

Another thing to consider in this scenario that hasn't been mentioned: you could be in violation of laws if you do leave the dog outside. Many areas have laws that make having an unrestrained animal a violation (even if it stays on private property). For instance, my neighborhood requires that dogs are behind a secure gate or on a leash if outside. Some areas also dictate how long a dog may be kept outside, or what must be available to them if they are outside. My city for example, require a dog is chained up outside for no more than 4 hours, and escape from direct sunlight, plus water must be available.

I say this because 1) you could be liable for similar rules and 2) it could also be a good way to let the owner know you are not able to comply with the request.

I don't know the right answer to your question, but is there a way you can rig up some shade for him that will give him a cool spot in your yard? I have also been known to put ice in water dishes.

If this is what the dog does at home, then I would leave him. He would be used to it and unaware of any potential issues. Bringing him in could disrupt the status quo at home. I would be more inclined to find an alternate for daytime, but have him sleep outside at night.