The rating you are referring to is the amount that Airstream retained for towing a car when fully loaded. It adds to the Combined carrying weight. That is the weight of the coach dry, all the liquids, water, gas, tanks, and the "stuff" you bring along on your trip.
You hit the answer in one regarding the brakes and tranny cooler. Add a tranny cooler, and get a Brake Buddy or somtimg like it and you should be fine with your honda. I towed a saturn sl2 behind our 76 and it was rated for 2K as well. I did have a trans cooler and a brake buddy. The unit will like it if you run with less stuff in it too.
BTW your honda weighs 2568 if it is a DX.
The tow bar issue is one that I have been thru. If you are looking to go cheap I would reccomend the RV rummage sales at some of the larger RV based communities in your area. I got a 4000LB Stomaster for $50.00 at one and it works fine. It is not the newest model, but it works. The baseplates are a diffrent story. If you are going used get the bar first then the baseplates. You may have to go new on the baseplates,I did.
The other option is to checkout www.hitchtrader.com
They have folks selling many different kinds of bars. Ebay works too, but do your research before you bid.
I would also check the owners manual of the car before towing it 4 down. Some of the hond manual transmissions were not all the manual and you could do damage.