As has been said, I have been towing for almost 4 years with an EV, and am now into my 5th season. I have just placed an order for my next tow car, which will be an EV (Polestar 4 to replace the Polestar 2).
As with all cars, you need to ensure it's suitable for your caravan (nose weight and towing weights observed). You will notice the reduced range when towing compared to solo (as with any car). My polestar 2 gets about 190 to 210 solo and 120 to 140 towing. I am expecting to achieve close to 180 with the Polestar 4 when towing.
Charging while towing typically means unhitching though there are locations that you can charge at when hitched. The new
Instavolt hub at Winchester is an example of a new location with such a capability.
There are a number of Facebook groups dedicated to EV caravanning, and if you have a car and caravan in mind there is a chance someone has some relevant experience to help you make a decision.
I certainly have no intention of returning to fossil fuel for towing. For me, the benefits of the electric drive train (torque, power, silence, response, efficiency, home charging, not having to visit fuel stations, lack of fumes and smell, and cost) massively out weigh the drawbacks of the need to unhitch when charging, which is perhaps 6 or 8 times a year depending on our trips.
If you are the kind of person who likes to hitch up and tow from the north of UK to the south of France, only stopping when you are totally out of diesel, then an EV may not be for you. If you are looking at an EV for other reasons (for me 90% of my miles are solo), and towing is just one minor (though important) use of the car, - and you have a home charging solution - then I would highly recommend it.
For charging when on site, CCC allow it. CMCC charge for it. Many CLs have warnings about not charging (as they are rightly concerned about being stung by the increased cost), but for those I have approached and asked permission with the intention to pay, they have been accepting. I carry a 3pin plug power meter with me that allows me to track usage and can report that usage to the site owners, so we can calculate costs.