While the heating issue is something to address (I have a laptop cooler under my UTVs), the only thing I can see this doing to the signal is changing the tolerance and therefore the output voltage of the UTV jacks which I could see confusing the multiswitch somewhat, but this is not something that I can see confusing another receiver while the UTV continues to get clean signal.
With ENORMOUS respect to lgodave (he's been here a long time and has helped more than his fair share of folks), i still gotta believe that an upgraded, powerered, multiswitch is the answer here. HOWEVER, as I'm not the one paying for said multiswitch, I do agree you should try EVERYTHING else first.
Good Luck! When (if) you do fix the problem, please post the solution here.