You 'know' that the van needs new tyres - how? There is no legal requirement for tyres to be changed on a caravan, so as long as they are legal - 1.6mm tread across the centre ¾ of the width and for the whole circumference - you are theoretically OK. However as the tyre spends most of its life standing still and under load there is a possibility that the tyre surface - particularly in the bottom of the tread grooves - could start to crack leading to delamination.
As there are no legal specs for the tyre per se the trade usually relies upon the age of the tyre and therein lies another quandary. A tyre life of 7 years always used to be the guide, but some (many?) dealers will now tell you 5 years and refuse to register your service if you don't agree to replacement (i.e. more profit for them). 5 years may be good advice if your van has budget tyres, but if you use good quality or even premium tyres I would suggest 7 years is probably safe. This of course assumes there are no other issues with the tyres such as potential blow-out lumps in the side walls, cuts anywhere on the surface, obvious uneven wear, air leaks etc., and that the caravan suspension is in good order.
It is your choice but I would suggest you wait until your MSE has done the service and see what he has to say about the tyres. If his advice is to change them (always change them in pairs) it gives you the opportunity to shop around for the most suitable tyre at the best price.
Per the comment in #3 above, I would doubt that the mover/tyre gap has ever been checked or adjusted since the mover was fitted. I know for a fact that none of mine have and my dealer is one of the best in the country! If the gap proves to be too tight (less than 20mm on a Powrtouch) with the new tyres fitted it is only two spanners and a bit of your time to adjust them yourself.*
(* Or if you have the tyres changed at home, a cup of good tea or coffee and some nice chocolate biscuits for the tyre fitter might get the job done for you?)