Bit the bullet today and signed up for TFL's automated tax system. It was working this time.
(My car is compliant with the letter of the law, but as a personal import it was rejected.)
Family run @12.50 GBP plus petrol/gasoline, plus 5 GBP drop-off is still cheaper than hiring a taxi to Heathrow. We have decided to try and do as many household chores as possible in a single day to spread out the hit to the wallet. Never drove her much anyway, but we like the option for emergencies.
The hidden problem is that when you consider their conditions and sign up to these things (on-street parking for example), they turn around and jack up the charges, repeatedly, which I did not sign up to.