The fee is to cover the grant of a license, not the checking of a person's suitability to possess a firearm. The right is a statutory one, the existing fee pays for the actual grant (printing the license, getting it signed, posting it to you). The public safety aspect/home visit, checking medical records etc is already within the standing remit of the police and thus comes from their general funding.
That is how the law was framed to fit our long existing constitution.
TBH I'm straining memories of a module on policing from the last millennium, however neither the original legislation that introduced licensing or the Bill of Rights 1689 have changed of course. Bill Harriman's book on shooting law gives a full account if one is very interested.