A quick sample of LA websites say 'no charge', as does this Govt document in section 9 but there appear to be some loopholes for the council to recover 'costs', which I assume are what's being used to screw you:
https://assets.publishing.service.g...ent_data/file/414670/definitive-map-guide.pdf
Not a process I've ever had to use myself, thank goodness. You can sometimes negotiate discretionary charges down by saying you consider X to be reasonable cost-recovery for them and any more than Y and you'll be requiring a detailed breakdown of the 'costs' with evidence that they were incurred and will be appealing anything that looks unreasonable. Council workers hate appeals because of the disruption and extra work they have to do so will often look favourably a way to avoid them.
On the other hand, there is a point of view that says that if you have a legal problem preventing sale of your house, £7.5k to make it go away is annoying but not as bad as it could be. As soon as it's resolved, the house will presumably become a bit more valuable to a buyer, which is presumably why the council feel entitled to charge you for their 'costs'.
Good luck, anyway