I had a very similar truck a few trucks ago :B.
I often towed with it flat to the floor for minutes at a time. Every time I hit a steep grade in fact. It would rev near redline until we crested. Never hurt a thing. On the flats it ran at right around 3000 RPM.
I was at or slightly over it's tow rating at the time though.
I never had any heat issues or break downs, but the 360 would burn a quart of oil every 500 miles when towing, but used none when not towing. My rv at the time was a 33' TT that weighed about 9000#.