Ah, so we're spending £2,000 to investigate an accusation of bullying that hasn't been made!
Not sure where to start with this. Firstly, the Council appears to have delegated its HR functions to a Staffing Committee (is it a committee or a sub-committee?), in which case, other councillors not on that committee have no role to play in relation to HR matters unless specifically called upon to do so by the members of that committee. If that committee is failing to perform its delegated duties, its membership should be reviewed at the Annual Meeting of the Council.
My advice, in the absence of a complaint from the member of staff concerned, would be to draw a line under everything that has gone before, consign it to history, and sort out your policies and procedures such that from this moment onwards, you become a model of efficient and compliant HR management. Don't poke the hornet's nest. If the staff are not complaining about bullying, why sow the seed in their minds? The model policies and procedures are out there, approved by the NALC and ACAS, adopt them, run a training session for all members on what they say and what they mean, set up the required structures and be confident in your ability to manage your staff professionally.