If the filer can prove that they speak French and have knowledge of French culture, they can gain French citizenship. It might require taking a few classes, but the most difficult hurdle (having lived there (legally?) for over five years) has been passed.

It'll take them more effort, but after living in France for long enough to lose UK voting rights I should hope that the rest of the citizenship requirements should be easy.

This assumes the filer wants French citizenship, obviously: if they want to stick with permanent residence but still want to vote then there's no solution, you can't have it both ways.