Most caravans that have a battery compartment fitted as standard will accept at least an 80 Ah leisure battery, and often a 110 or 120 Ah leisure battery. Either battery when properly charged should successfully operate a caravan mover.
However. Whether you need such a big battery will depend on how you intend to caravan. For example. If you want to caravan without access to mains power, then you will need the biggest leisure battery you can fit. But if you only ever use pitches with mains power then you could get away with a much smaller standard car battery just to run the caravan mover.
It often comes as quite a surprise how little total power a caravan mover needs for example most caravanner's typically use less than 2Ah of battery charge to position a caravan on site, or into storage. This is because the motors are not running for very long. But whilst the total power used is surprisingly small, the motors on some movers do need up to 40A each (but most are closer to a peak current of about 20A each) as motors begin to turn. The battery needs to have a big enough starting or cranking current to supply the peak current demands. As soon as the motors are turning their current demands drop significantly, so you don't actually need a battery with a big capacity.
If your not sure about your type of usage, your better to hedge your bets and go for the largest leisure battery that will fit in your battery box.