The car can do its part just fine so long as you do yours.
I hauled a 3600# total weight dual axle box trailer for a trip that was over 1000 miles, part of which was through the mountains of Virginia, part of which was heavy rain/storming, bits of stop and go traffic in the peak of summer, etc.
I also regularly haul my brother's 2000 lb fiberglass boat since he no longer has a truck of his own. We have no problem filling the fuel tank of that boat full, throwing a couple coolers and gear in, and driving the whole thing to the lake with 5 people in the car to boot. Launching and pulling the boat back out of the water has been no issue.
I haul utility trailers loaded around the 2000 lb mark pretty often in the summer.
I'm not saying it will do it easily, however. Whether or not it will work OK depends a lot on you. Any moron can buy a dodge ram and hook a boat up to it and drive around just about exactly the same as he does when he has no trailer, and the truck pretty much takes care of everything via brute-force and the driver needs no real special skill or thought put into what he is doing. This is not the case with the Compass. Its totally fine to operate the Compass around its limits as long as you do it with thought and care.
The requirements, according to the manual:
1) 9 speed auto with 4x4
2) Use Autostick / Manual shifting when towing heavy loads. Yes, this is written in the book. The 9 speed has four overdrives, you will likely use none of them ever when towing 2000+ pounds. That is to say, you will never get beyond 5th gear on flat ground on a nice day, maybe 4th in hilly terrain depending on your speed. On a long downhill stretch on the highway you might visit 6th briefly but probably not. This 4 cylinder simply doesn't have the torque for it. FWIW I don't use autostick much when towing empty trailers or loads under 1000 lbs. The instant MPG gauge can be referenced to help infer overall engine load but you also have to just "feel it out" somewhat. Its OK to keep the RPMs up, thats where the power is, its much less stressful for the engine to turn 3500 RPMs or even more going uphill with a 2000 lb load than to be lugging under 2k.
The additional requirements, according to me:
3) Factory tow package. For super light aluminum boats, bike racks, probably even light trailers with a single 4 wheeler, etc I think you can get away with an aftermarket towing receiver but its best to have a Compass that came with the tow package right from the factory if you are going to tow at 2000+ lbs. The height of the factory receiver is "proper", allowing a normal drop hitch with ample room for the chains to do their job if they have to. Aftermarket receivers mounted under the bumper are too close to the ground for the chains to cradle the tongue if the hitch fails. Don't forget to cross your chains. The other thing is the factory tow package also includes anti-trailer-sway programming in the ABS and you definitely want that.
4) Trailer brakes, if at all possible. Despite its puny engine power there is enough gear leverage to get a tremendously heavy load rolling. Stopping it safely is just as important. The 3600 lb trailer I hauled had a surge brake that worked very well in emergency stops and behaved very well in the mountains. Other trailers I use like my brother's boat trailer are equipped with electric brakes. There is no factory brake controller option for the Compass but a lot of inexpensive aftermarket ones are available. You can slide by without this if say its a short drive to the lake and the roads are maybe max 55 MPH or something, but still need to be careful.
It goes without saying that you need to be sure your car's brakes are in excellent condition, tire pressures correct and all that. Another nice-to-have is an anti-rattle hitch plate. Don't be cheap when you buy your ball and drop hitch. Be careful driving around and you should have no problems like me. I've never had any overheating issues or warning lights of any kind during heavy towing, not even on that long and grossly overloaded summer trip, I think 226 degrees is the max I ever saw during the stop-and-go traffic.
Be safe and have fun.