The requirement in your code of conduct is there to protect the financial interests of the council, for employees it is contractual (under GDPR) therefore allowed under GDPR as long as they are informed when they sign up.
Councillors do not have a contract, per se, but they in undertaking a financial management role, and are under a legal obligation to act in the best interests of the council, this is the 'statutory' item in GDPR, as long as they know that as a councillor on a particular committee the check will be done, why and who will have access to the records and how long they will maintained / updated then it is legal.
GDPR doesn't prevent the these types of check, you just have to ensure everyone knows they exist, why, how the results will be used and how long they are kept for.