Subsidence with the known cause being a broken drain should not affect your future prospects of getting insurance. Once the drain is fixed the problem is not going to re occur. When I worked in Shoeburyness my landlady had the same problem as you caused by her own drain and only picked up when the back of the house began to move away from the front. She had no issues with continuing insurance.
It does seem that only a solicitors letter is going to focus your neighbours mind on this. Again as I said before there is no need to fall out over it you are sending him the letter to help him in dealing with his insurance. However if he doesn’t pursue it with them it’s time to sue him. He may be able to fix his own drain but your not going to leave him to underpin your house.
“Speak softly and carry a big stick”