I guess the question is do you really want to end up with a bike trailer, or is it going to be something you'll need twice and never use again. If the latter, you can rent bike trailers from some truck rental places that are purpose made for doing just that. Or I guess you could rent a pickup or small box truck for a one way move. Would probably be quite a bit cheaper for something you're only going to ever use a few times, and then you don't have to worry about storing it either...
Also, 1000 miles is a pretty decent distance. I think that for traveling any appreciable distance, hauling a trailer would be pretty unpleasant. You're limited in your top speed, and your turning abilities are severely compromised. Not to mention constantly worrying about whether your trailer is still attached. :-D A pickup might be better suited to the job, assuming you don't care about having your car while you're there.
Or, you could try something like this:
http://www.versahaul.com/, but I suspect that unless you're driving something pretty beefy, you probably won't be able to fit a Sprint on it...
Good luck!