The end:
Having made no progress with the local main dealer or the manufacturer's 'customer services' department, my lass sent an email to the 'customer services' department of the manufacturer's parent company, it was ignored. A second email was sent, it too was ignored.
The car was taken to a local garage where the problem was immediately identified as leaky thermostat housing, and she was told it's a known problem with the make and model of car in question (subsequent Google research suggests that's true).
A new t-stat assembly was fitted and all is now good, so just a few points remain:
If, as the local garage suggests, the problem is a common one, I assume the main dealer didn't want to acknowledge it at the outset as it would hit their warranty costs, so instead they just demanded £125 + VAT to investigate something that they had already declared to be outside the scope of warranty - without knowing the cause of the problem.
The manufacturer's customer services department was happy with the main dealer's stance.
Customer services within the manufacturer's parent company did not reply to emails sent using their own customer services link.
Now, the problem was fixed by the local garage for an all in price of £90 (inc parts and VAT).
With an hourly rate of £125 the main dealer would probably have chalked up charges in the order of £250 + parts and VAT for the same work.
As it happens I was recently considering a car from another company within the same group, but given my daughter's recent experience, and ongoing discussions about parts availability in yet another of their brands, I'd be inclined to give Stellantis a wide berth in future.