It's possible to rent these cameras at some home improvement big box stores. Since they are rather spendy, and you'd really only want them for a day or two, that might be the way to go.
The tires are always going to be warm, that's where all the energy from the rolling resistance is going. Even at a real high psi, the lugs are going to wiggle around and get warm, especially with the soft compounded siped snow tires like OP has.
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