I just have a 3/4 ton. I haul around 10,000# on a regular basis, and have hauled up to somewhere between 14,000 and 15,000 on rare occasions. I can tell you with 100% certainty that if I hauled any more weight than that 10,000 on any kind of regular interval I would have a DRW 1 ton. My friend let me haul a lighter load, probably 7,000-8,000 lbs with his dually on a 250 mile out and back trip. It was a very noticeable difference. Not as much difference as 1/2 ton to 3/4 ton though.
I'm not sure how much squat would be different on a 1 ton SRW vs DRW. In my 3/4 ton I have to be a lot more careful of tongue weight than the guys with the DRW trucks. But, that is not the only difference by far. In my experience my SRW truck requires more active driving than the DRW truck. Meaning, the SRW seems to "wander" a little bit more on the road, where the DRW seemed to hold a line without much input at all, again not massive but definitely noticeable. Also there is a lot less movement when taking turns and curves. The SRW seems to have less lateral "hold" and the trailer can feel like it pushes the rear of the truck side to side. Not sliding the tires on the road, but flexing the sidewalls quite a bit side to side like if you were to try to balance on a beachball. That was noticeably less with the DRW.
As for tight spaces, I don't think the difference is that great. they are both going to be large, long trucks. With either one you aren't going to park up close to the store in a crowded lot.