Were you tested in France? If so, do you have a copy of the test result? You were given the standard dose which is the same as that recommended in NICE guidelines. However, doxy is normally prescribed in the first instance but it can be brutal.
Do look at “Ongoing symptoms after a course of antibiotics” in this NICE link:
https://www.nice.org.uk/guidance/ng95/chapter/Recommendations#diagnosis
Royal College of General Practitioners Lyme disease toolkit:
https://elearning.rcgp.org.uk/mod/book/view.php?id=12535&chapterid=408
Whichever way you look at it, you have ongoing symptoms and the guideline allows for a further course of antibiotics. If I were in your situation then I’d be grabbing the doxy and preparing for a rough ride for a few weeks. All the NHS can offer is 2 courses of antibiotics, they have no Lyme experts and you’ll be left on your own to figure it meaning private treatment which ain’t cheap.
Good luck with navigating this and do shout out if you have any q’s.