I've got a similar-sized SUV [Montero Sport]. I'd consider putting on a front end 2" Receiver and buy or custom-fab a front Cargo holder for the moderate-size Genny you're considering. It wouldn't affect cooling or Headlights, and would balance SUV loading w/o putting great weight on the front Suspension [<120 lbs.?]. Certainly not the weight of a Snow Plow, and probably not too different than the Brush Bar some folks retrofit on for looks. The Genny weight might even provide some Tongue weight equalization benefit...
One quickie Ratchet Strap, or alt. method, could hold it down. A mod'ded or homemade cover would keep off Road grime. A Bike Security Cable could lock it to your frame. With a Receiver Pin Lock, all would be secured. Unload the Genny at your leisure.
With +12 VDC right there up front anyway, you might be able to pump Gas with an Electric Pump to storage that then feeds by Gravity Siphon to a Genny Fuel Line 'tee'. LOTS of Toy Hauler Trailer Owners safely modify Fuel pumping functions off of larger Tanks to gas up their Toys. See write-ups on line. But, I have no interest in starting a safety debate when you're the sole User.
You could pull into Camp and fire up the Genny where it sat, if you like, with a bit of noise immunity due to Genny position vis-a-vis your TT.
Due to 'excess' Tongue Weight on my new/used TT, I'll be hauling my much-heavier Genny inside, strapped down well, just in back of the TT Axle. We all haul around exposed Propane Tanks and spare Gas sometimes. So, I'm not inordinately worried about a Gennie in my TT.