I think you have two viable lines of attack. Possibly the simplest is a Freedom of Information request. You can read guidance from the Information Commissioner at https://ico.org.uk/media/1187/section_40_requests_for_personal_data_about_employees.pdf
. The conclusion there is: "However, even though a job description may constitute personal data in these circumstances, it is likely that it would be fair to release it in response to a FOIA request".
If a council document is available through a FoI request, then certainly it should be a fortiori available to councillors without demur.
The other line of attack is that access is required to do the job of councillor. This is a common law principle. If the council has a staffing committee, then members of that committee should certainly be granted access to all relevant employment documents. If there is no staffing committee (or equivalent) then all councillors should be granted access, since the council must collectively fulfill the role of employer. The argument for a non-member of the staffing committee is weaker, but if it can be shown that there are issues where all councillors should understand the employment arrangements, then this argument can still be made.