The decision rests on whether you intend to tow east of the Rockies or out here in the mountain and Pacific west. The 2013 F250/350 Fords offer a gasser that will handle your trailer's 12100-lb GVWR. Out west only the diesel will do the job.
The critical element is altitude. On the east side of the country, that Cougar wants a tow vehicle with a tow rating of 13,000 lbs. That allows for truck loads beyond a 154-lb driver and fuel.
Out in the west with a gasoline engine, altitude lowers the output of unturbo-ed gas engines by 3% of rated HP per 1000 feet. The Cougar would need a gasoline-engined truck with a tow rating of 14,520 lbs !
You really want a diesel with that big 5er.