As you say, it will be best to seek agreement among the parties. Certainly, if the clerk refuses to carry out essential parts of the job and is preventing the council from operating normally, that would be grounds for dismissal. Although, as always with employment matters, the situation must be handled carefully with full documentation of every step. Given the very high potential cost of employment disputes, it may well be worth seeking professional advice from a lawyer specialising in employment matters.
A council cannot function with no proper officer at all. So it is essential for the council to appoint someone, even if they only carry out a minimum of duties. Usually a councillor takes on the role temporarily, but they cannot be paid. If the council wishes to appoint a new clerk, then the search for candidates can use any effective means, provided fair opportunities are offered to all prospective candidates.
If a council is unable to operate, then, as you say, the District Council has reserve powers to intervene.