I would bet money that 50% of your sway issues are the C rated tires. The sidewall and carcass construction of the Raptor's specific BFGs are designed for small bump compliance and ride comfort through the rough stuff.
If you don't believe that the difference matters, take a look at the PSI rating for a C rated tire versus an E rated tire ( like Toyo MTS). The higher the PSI capability, the less sway you'll encounter and the more planted you'll feel. The higher load rating also means that the effect of heat at higher speeds is significantly diminished.
The tires also have a lot to do with the diminished tow capacity over F-150s.
If I was going to tow that tractor with a raptor, I'd want to massage the raptor with the following modifications:
1 - D or E rated tires.
2 - class 4 hitch with WD
3 - a trailer with brakes on both axles
4 - Deaver HD leaf springs
5 - trailer tire TPMS
As far as the trailer with the tractor on it, you have to know that you probably won't get the balance of the trailer/tractor perfect. you'll end up with either a little more tongue weight or a little less than optimal. This means that the Tires on the trailer aren't going to heat evenly. To combat the product of uneven load, consider putting F or G rated trailer tires on.
No one really thinks about it until they have a trailer blowout at speed. I never gave it any thought until I had an E rated trailer tire delaminate and blow out while doing 75mph at 1AM in the middle of nowhere. The trailer tires had less than 7K miles on them at the time.
For reference, this was towing my raptor ( I'd guess that it is 6300 lbs now) on a 20 ft, 14K lb capable drive-over-fender trailer that weighs 3400 lbs unloaded. ( which I prefer to keep the COG lower and the trailer sway down)
you will also want to make sure that you test and tweak the settings on your trailer brake controller. It will be hugely important to get this right so the trailer and truck together stop "flat". At this close to the weight limit of the raptor, the last thing you want to deal with is the trailer bucking during emergency maneuvers due to the trailer not slowing as fast as the truck, or the trailer slowing faster than the truck.
hope this helps.