...yes there is evidence as everything was discussed on a Teams call that they recorded.

To go a little deeper into the issue they've got...when I bought the car last summer it was explained to me that it had been bought from the MOD (or from a lease company they used) and therefore there was a need to re-register it under a different registration plate. I was given all the paperwork relating to that - but it meant that the V5 document showed the date of first registration was 2019 but the date of first registration in the UK was 2024 (when the garage bought it from the MOD).

The sales guy's manager called me to explain what had happened. They are saying that because of the re-registration, when they provided the price for the part-exchange they had mistakenly looked up the book price for a 2024 car rather than a 2019 car, however, when they couldn't find the new registration plate on one of their systems, they looked up the old registration (which I had provided) and realised that they had made a mistake.
They then looked up the correct book price for a 2019 car (about 70% of the price they had given me) and THEN said that they also needed to cut that number to 60% purely because it used to be an MOD car and they are "more difficult to sell" apparently - so that took it down to around half of the part-ex price that they have provided and done the deal at.

The current situation is that I've thanked him for explaining what has happened and that I just need to take some time to process it all and think about what I want to do next as I really want the car.

There are some bits of legalese in the documents that have been digitally signed - I can post them in here if it might be useful.