I think that would be unadvisable.
The reason is, even though some of the trailers were supposed to weigh "995 pounds" from the factory, I'm not sure that was ever true. And even if it was, that was without such "extras" as
propane tanks, spare
tires, dishes, clothing, refrigerators, water, etc.
I think that even packing
light and watching your
weight, you'd be up close to 1200# minimum. And percentage-wise, even 1100# would be 10% over your tow rating.
Also, there is a chance that your car's rating expects you to subtract for passengers or cargo in the car itself --- that's not always the case, but bears checking.
The lightest egg I can think of would be a Compact Jr. type, but even there I think you'd be hard-pressed to stay under 1000#, all up.
I can specifically remember one of our members, Gina D., talking about towing a
Burro with her Element, rated for 1500#. With trailer
brakes she felt it was fine, but...
just. I doubt she would have wanted a vehicle with 33% lower tow rating (not to speak for her, but she did have a Burro - she did pack things along that one might leave at home to save
weight, but then her Burro also had an unusual aluminum frame, so that was lighter).
I don't mean to rain on your parade, but 1000# seems like it would direct you to a teardrop (the ones that have a bed inside and then a kitchen out back). I don't know their
weight, but probably closer to 600# or so?
A teardrop would also have less wind resistance. That would be another factor when towing a "high" egg behind a small car.
Or.... another tow vehicle and an egg