For the sum involved use small claims, you then have protection from having to pay their costs if you fail. And you can't get to the high court without first starting at a lower court and then either party appealing the decision and an appeal being allowed.
This sometimes does work out in a more straightforward manner... I asked Cerberus for an FTTPoD desk survey in May 2020. It came back a week later as 'unable to quote... FTTP is already planned for this location'.
The exchange appeared in the Openreach build programme document a few months later, and was completed around August 2021.
Anyway, good result here. You'll probably find an altnet is hot on Openreach's heels - that was what happened to me. Waiting for FTTP in some places is like waiting for a bus... two turn up at once.