Resolution
The fitter tells me that the only conclusion he can come to is that (like some VW models) the Seat has a cut-out function if the towing electrics are left unused for a while (and I don't tow very often), which cuts out the live feed to the towing outputs as they relate to the lights - he says that the other feed, which runs things like the internal electrics in the caravan (such as the fridge) remains live. As he put it, you can keep your beer cold but don't have any running lights on the towed item!
The solution is to turn off the ignition and remove the two (red) fuses for the rear lighting circuit (which I believe are numbered 40 and 41 in the Seat manual), which are at the top left of the fuse box under the steering wheel, leave them out for 20 seconds or more, then re-insert the ones you removed - note that there is no need to replace the fuses. Turn on the ignition and the outlet to the towed vehicle should be live. Sometimes, he says, you may need to repeat the process and also remove the yellow fuse immediately to the left of the two red ones, then re-insert all of them and have a live outlet.
Seems whacky to me, but it works!
Logically, the only place this cut-off
can operate is in a relay between the rear light clusters (from which the feed to the towing outlets comes) and the towing outlets - as otherwise, surely the car would drop the power to the rear light clusters. I neglected to ask him whether the relay was native to the Seat or was part of the kit he installed.
So, I now know
how to resolve this issue, and hope the note is helpful to others.
It would be interesting to know if anyone knows if the relay is integral on an 07 Seat Altea XL, or would have to be fitted on installing the towbar and outlets.