Does anyone know if the finance company bear any liability or clout to get this fixed?
By all means, double check this as I left the finance world more than 10 years ago, but a bit like credit card purchases, the finance company effectively provide the car to your daughter until it’s paid for. That’s why finance companies will only finance cars from ‘reputable dealers’.
That said, it doesn’t mean they are any more liable for faults than the dealership would be.
Based on your post, I don’t think they’ll be much help, hindsight is great, but if the head gasket failed within 12 months of buying it, that was the time to go back to the dealer to complain and if they blanked you, got the finance co involved, now 2.5 years after purchase there’s no liability for them really.
It sounds to me there’s a fundamental problem with it, a 5 year old car really shouldn’t need a head gasket, that said, if the water pump has failed (which on some cars should be replaced at certain internals, usually at the same time as the timing belt) it could cause a head gasket failure due to over-heating.
I think your daughter best bet to get out of it is to VT, or Voluntary Terminate, in order to do that she needs to have paid 50% of the total finance cost of the vehicle. Her finance documents should tell her how much she will have needed to pay to reach that point, that figure includes any deposit she paid, and importantly any final payment.
If she’s able to VT, she should take the time to consider the pitfalls of that, it’s a very grey area.