"The unit is nice and level." The appearance of the caravan being level is not a requirement for it to be at the correct towing height, its just a pleasing aesthetic if it works out that way.
The key to the correct towing height is set out in EU legislation that your car should have been subject to, which requires the LOADED tow ball to settle between 350 and 420 mm vertically from the ground to the geometric centre of the tow ball.
On the basis that you must also have a nose load that presses down on the tow ball, when the caravan is hitched the tow ball will be depressed from its unloaded state.
You tell us your measured height is only 345 mm but you don't tell us if that was the loaded or unloaded height, but in either case it is less than the EU regulations permit.
A number of things may have gone wrong.
Some sports versions of standard cars are not always permitted to tow, You will need to check your VIN weight plate to see if there is Gross Train Weight figure. If there is none, and you tow you are actually breaking the law, as the car will automatically be deemed to be overloaded and thus legally unsafe.
It's possible the a towbar for the standard vehicle has been erroneously fitted fitted to your car, which because of its lowered suspension, the tow ball is lower than it should be.
Others will disagree with my opinion on this point but No vehicle needs spring assisters. They are not a solution to overloading or to correct other faults with the vehicles suspension. On the basis that you do not overload the hitch or the boot, the vehicles suspension is designed to carry the maximum load specified for the vehicle, and that should ensure the tow ball falls into the correct height range. If its not in range, then it could mean you have worn components in the cars suspension and that should be put right rather than masked by adding assisters.
Some more modern cars have ride height sensors that feed into the stability control systems. If you add assisters to these vehicles, the suspension will be sending false readings to the stability control system, which will means it gives incorrect correction outputs. That could make it less stable rather than improving matters.
Caravan manufacturers are fully aware of the tow hitch ride height range, and they will have engineered their jockey wheel design to provide enough ground clearance for normal road conditions. Its possible a previous owner may have fitted a larger jockey wheel, but normally they will retract so the axle of the wheel sits just above the bottom of the a frame.