I agree with you about a car battery coping with the demands of a mover, but they do not like modest discharge currents over long periods as you might require to run a caravan where there is no mains hook up available.
Where I disagree with you is that "leisure" batteries are modern name for "Traction" batteries. These batteries are designed for battery powered vehicles, (milk floats, fork lift tucks, motorised wheelchair and invalid carriages). They can cope with the peak demands of a caravan mover which last for a fraction of a second whilst the motors start to spin. Under normal running conditions the motors typically only draw a maximum of about 25A under heavy load. These magnitudes of current are within the capabilities of a properly maintained leisure/traction battery, and the battery should be able to go to provide adequate power for the rest of the caravan for a few days at least.