You’re right, moving companies do this, but they generally aren’t big fans of it, and they’re very unlikely to do it in the summer months when they have a lot of demand for business.

Towing a car is no problem. I don’t know the rental company’s policy on having a car inside the truck, but I’m betting they forbid it – getting a car into a truck is no small feat, and the risk for damage to the car, or injury to people loading it, would probably lead them to forbid it.  There’s also the danger of having the car inside the truck – shifting around, gas fumes, etc.

Still, doesn’t hurt to ask. You also have the option of shipping the car separately -- this article helps explain how to ship a car. If you don't want to do that, maybe a friend would be willing to drive the second car?