you might consider getting a winch with a contactor, instead of a solenoid. they are much smaller. or, get a contactor for whatever winch you choose, if it doesn't already come with one.
it would be much easier to find a space to mount it, or even mount it in a smaller box. my winch uses a contactor, the winch control box is about 5x larger than it needs to be. its not an issue where mine I mounted, so I just use the original configuration.
edit: I just checked that link. that is a contactor, not solenoid. though, my concern about the heat remains.