You could do it but you'll be at the limits and likely over the tongue rating. Sway control and WD hitches as mentioned above are 2 totally different things, although some WD hitches have sway control built into the design. The T4R manual only mentions a sway device because you can't use a WDH with the stock receiver. The OEM receiver is mounted to the crossmember and is not designed to take the stress of a WDH. You can twist the frame and it's not repairable. You will need an aftermarket receiver that is frame mounted but unless it has changed recently nobody makes one.
I had a 4th gen V6 and towed a 19' hybrid. It weighed 3800lbs loaded with under 500lbs of tongue weight. Plenty of engine but struggled a bit on the highways because of the 4 speed trans and gearing. The 5th gen has more power and a 5 speed which would do a better job. I did use a WDH with the OEM receiver as well with no problems, didn't know about the hitch restriction at the time. I upgraded to a V8 and it towed like it wasn't there. Eventually got a bigger trailer (26'BH at 5600lbs). Plenty of power but ride got a bit squirelly due to the length vs short wheelbase.
As mentioned, you would probably be fine with that trailer but it really depends on where you tow, how often, how far etc. Mainly flat, short trips, no problem. Long trips or in the mountains, you may want to find something a little lighter. My '06 Starcraft Antigua 195ck was 2800lbs dry for example.