No Worries Jules, no offence taken. I skimmed the post quickly and missed the bit about the extension so my oversight really, apologies to the OP.
I wasn't really suggesting cutting roots or causing wilful damage but if the subsidence reoccurred and was shown on the balance of probabilities to be down to the tree then mitigation works would be required whatever they may be, this would be exempt in my view. That is realistically topping or felling, root pruning is a waste of time as the use of root barriers in not recommended by NHBC and the roots will grow back as you say. I still think TPOing a tree with low visibility on a shrinkable soil and adjacent to a 1930s house (with previous subs issues) is just bonkers.
There was a claim a few years ago (I forget the ref) where the applicant submitted evidence showing seasonal movement and the LPA refused inviting them the appeal. They took some legal advice and did not bother with the appeal and instead went straight to putting in a claim against the LPA. The LPA used the defence that the appeal route was available. The applicant won the cost of underpinning as the court took the view that the LPA decision was wrong irrespective of the appeal. Basically why should they mess about appealing the LPAs incorrect decision while damage is clearly occurring. I would personally always go for appeal but the PINS can take up to 27 weeks to determine an appeal so that can be an issue if damage is on going so I kind of see the logic. It would probably be worth starting the ball rolling on both.
My main point though was get some appropriate professional advice on trees, soils, the underpinning, and the value of the property and how it will be impacted by the underpinning. Its not a straight forward purchase.
Cheers