Difficult issue. But I would think it would certainly be better for the councillor in question to keep away from the matter, if they have made it clear that they already have a settled opinion.
Staffing questions need to be handled carefully and decisions must be made on relevant considerations; redundancy requires consultation and that the job is truly redundant. Claims of insubordination do not seem relevant to whether a job is redundant, or who should be made redundant.
A member who has stated a position before any of the procedures have started is liable to put the council in a difficult position, especially if the member of staff takes the matter to a tribunal. Badly handled termination of employment can cost a council a substantial sum. So, on the stated facts, it would seem best for the councillor to withdraw, or for the council to choose not to appoint them to the staffing committee.