I would let it go. Life is too short, as evidenced by a guy dying before his instalment plan finished.
You have no proof of this verbal contract, you have no proof that the payment by 4 instalments deal wasn't a payment by 3 instalments deal and the woman just lost her husband, who was supposedly a good friend of your dad.
I am sure you wouldn't appreciate a good friend of your dad bothering your mum about something like this if he died?
The right thing to do here is sign the transfer form (to get it out of your dad's name as mentioned by others) and tell your dad in future to make paper agreements even with good friends.
