I love my 2013 Prius V so much I don't want to sell it But am wanting to tow a light weight camper of 1000 pounds. Is there any way this is possible without hurting my favorite car?
If you are strictly going by what the manufacturer states as a towing capacity for a vehicle, your vehicle wasn't designed with towing in mind. Looking at this page states the 2016 was the first Prius model made with towing in mind. It can tow a little over 1500lbs.
If you want to follow Toyota manufacturing spec, you should not tow with your car. However, I have also read plenty of forum entries which state owners have had no issues towing small trailers with their Prius vehicles. You will have to decide at your own risk if this is something you want to have happen.
The payload rating for the 2003 Toyota Prius V is 1047 lbs, so in theory 1000 lbs of extra weight is within tolerance of this vehicles payload. On the other hand, Toyota recommends that you do not tow anything with this vehicle, as it is not designed to do so.
Personally, I would say that since the majority of the campers weight is on its own axle, you won't be overloading the rear suspension, and as long as you are careful not to stray too far over that 1000 lb limit (keeping in mind that is the payload rating, this car was NOT designed or rated to tow ANYTHING), it shouldn't damage your cars drivetrain or chassis beyond normal wear and tear.
Just be sure to take it slow! Good luck!
If you're going to ignore Toyota's specs - and you might reasonably do so, not rating the car for towing could have been based on any number of considerations quite separate from engineering - there a few things you should take it account:
The payload of the vehicle includes the tongue weight of the trailer along with any passengers and cargo in the car.
Your brakes will be stopping both the car and the trailer. A car rated for towing would have brake capacity to match the spec for the trailer weight. A car that is not rated for towing might not. You'll only know that in extreme circumstances. While the car's suspension only has to handle the tongue weight of the trailer the brakes have to handle the full weight of the trailer.
The drive train and cooling will also be designed to the load of the car without the trailer, so there may not be sufficient additional capacity for "extreme" conditions. Again this is a judgement call, if you don't climb steep hills at altitude on hot days it may not matter at all.
A car that is designed to tow has at least been thought about with these considerations and the load of a trailer in mind. Your car may not have been. It almost certainly has adequate capacity to handle towing when the overall load is within limits and the conditions are moderate, but it may not perform in extremes. The place where I would be most concerned is with the brakes.
On the other hand, the car may be built with components that are shared with other – larger, heavier – cars and so you may have adequate braking capacity, not because the car was engineered for it, but because it was simpler that way.