Hello Philip,
As you can see from the above replies. No one has used a jockey wheel or 'A frame' mover on the TA's. With the exception of Michael E (who does not use a mover at all) everyone else has gone to the movers that drive the main caravan wheels.
As you know moving a TA is simple enough in a straight line, but when attempting to turn a corner, the main wheels need to take a different track, but they are restrained by the axles, so at least two tyres end up scrubbing and making the manoeuvre difficult and hard work. The same is still true for the mover, and even the best still can't turn on a sixpence like a single axle van can.
The same is true for the A frame movers. It is often necessary to shuffle round a tight bend until the full turn has been achieved. Turning a TA needs a lot of power.
Movers will take effort out the manually positioning them, which is why they are so popular.
Unfortunately there are some significant performance and safety differences between A frame movers and main wheel movers. This is nothing to do with the caravan beings TA or a SA but simply the mass of the caravan
The fundamental safety issue is the ability of the mover to stop the caravan on a slope. If you cannot stop a caravan on a slope than you have a lethal device, that will crush, maim and even kill.
All movers rely on the grip between the driven wheel and the surface on which they sit. A hard dry surface is usually the best, but anything that degrades the grip will increase the danger.
The main difference between A frame and axle movers is the amount of grip and therefore control they can muster. The amount of grip is determined by how much force is pushing the wheel to the ground. For the main wheels you have the whole weight of the caravan, but for an A frame mover it is the weight of the mover plus just the nose weight of the caravan.
As a guide, if the caravan cannot be controlled by one person manually, then it is unlikely that an 'A frame' mover will offer much if any additional control and thus safety. By comparison, the main wheel movers are roughly the equivalent of 10 people at least.
If you only ever want to use the mover on level firm ground, then an 'A frame' unit will probably do very well (such as a caravan dealership), but if you want the mover to work in real life situations like on slopes, or grass land, then the more universal units are the main wheel movers.
Michael says that TA's are easier to reverse. That is probably due to the length of the caravan, rather than the fact it has twin axles. Longer trailers are easier to reverse, because the direction of trailer changes less rapidly to the same steering input.