Since the Camry is designed to carry 5 adults and their luggage, towing 1000lbs isn't going to hurt it anymore than loading the car up with your friends for a weekend vacation. 5 adults + luggage can easily surpass 1000lbs. The Camry has a 1000lb payload capacity, as well as a 1000lb tow capacity (However, if you are towing 1000lbs, you can't have any payload, and vise versa). Keep in mind that if you add weight to one, you have to subtract from the other. IE, if you have 500lbs of people and cargo, you can only tow 500lbs according to Toyota.
Towing 1000lbs vs having a 1000lb payload is actually better for the car in regards to wear on the tires and suspension due to the majority of the weight being upheld by the trailer and only 100lbs on the tongue, which is like having a couple large bags of dog food in the trunk. Has your suspension ever collapse from putting 2 bags of dog food in your trunk? In regards to the brakes, engine, and transmission, both will have equal wear on those parts since the 1000lbs is the same to those components no matter where the weight is located.
I find it surprising that a mechanic would say that a car designed to carry 5 adults + luggage, which could easily surpass 1000lbs, doesn't have the power to tow 1000lbs. 1000lbs is 1000lbs.