The 1,560lbs towing capacity is the LEGAL limit, it can not be changed by anything, it is tied to the VIN, so it is set in stone.
If you tow above vehicle capacity and there is an accident your insurance company "may" walk away, not for sure, but legally they have an "out", accident doesn't have to be weight related.
So that's the legal side of it, the "may" you tow a 2,000lbs trailer, answer is no.
"Can" you tow a 2,000lbs trailer, yes
You will probably need a new clutch after the trip, 2.5ls don't have alot of torque, lol.
And for sure hook up trailer brakes, that 2,000lbs will push you around quite a bit, the above axle mounting(5th wheel) does help, but the regular cab Ranger is only 3,000lbs soaking wet.
If you take it easy and don't expect to be zooming up even moderate hills you should be OK.