I'm not going to be much help with your query but just wanted to say that your circumstances are just as it happened in my case last week.
Having had a look at the abundance of data on the internet, I must admit that I've not reached a decision that I could say that I'm confident with. However, what I feel strongly is by not doing anything i.e. ignoring, PE is most likely to continue and harass with more unsavory letters but at not much real cost for them. Whereas if I appeal their charge aka. "the speculative invoice" then they are very likely to reject it giving me the opportunity to take the POPLA route. I understand that the POPLA appeal would cost PE money just for the appeal to be considered.
In the event where POPLA appeal fails then I plan to switch to ignore mode but what I am unsure is
1. will I complicate matters by engaging with PE in the first place?
2. albeit being a "private" dispute, will I weaken my position by not engaging or following the appeal process even though it may be inherently flawed or unjustly executed (due to my belief that PE is not in a position to charge for the "overstay" let alone then intimidate by sending an invoice for a service that is neither quantified nor performed correctly.
To do or not to do is the quandry I'm stuck in
Good luck and hopefully you'll come to a decision much quicker than I have!