The analysis is different for law firms and lawyers, who have an ethical obligation to charge "reasonable fees" (in the U.S. pursuant to various versions of Rule of Professional Conduct 1.5) and for other businesses, which are not subject to the same regulation.
While my personal business practice has always been not to bill clients for discussions about billing, and that is a common business practice among lawyers, there is not a per se prohibition against doing so. It would usually be considered reasonable to charge fees involved in doing the same work in connection with a court filing such as a bill of costs or other fee request from a court. In either case, for this kind of work, reasonable fee requirements might end up requiring that this work be done by a paralegal or other administrative staff person rather than a lawyer charging a full legal fee rate, when possible.
Whether these charges are authorized would also depend upon whether the contract between the parties authorized them, particularly in the case of a non-lawyer party where contract authorization is usually the dominant consideration regarding whether charges are authorized.