OP,
While I agree you won't like towing with that little engine no one bothered to point out your weight math was wrong. You were subtracting the vehicle's GVWR from it's GCWR to find what you could tow. This generally isn't correct. You would need to weigh the vehicle loaded with your family and gear to see what it weighs (likely well under the GVWR). You would subtract it's scale weight from the GCWR to determine how much you could tow. You would also need to make sure you have enough weigh capacity (GVWR - scale weight) for the tongue weight of your trailer (generally about 13% of loaded weight).