Courts are very familiar with adjudicating such ambiguity and have been throughout history. They simply have to avoid discriminating in ways that violate the law, but other forms of subjectivity.

As a stupid but illustrative hypothetical example, if the official decides each morning whether he will approve all CCW applications he sees that day or none of them based on a die roll before leaving home, and then unilaterally changes it a month later to go based on whether the last digit of the local CBS affiliate's high temperature forecast from the 7am newscast is odd or even, these stupid rules and rule changes would be completely compliant with the equal protection clause.

They have to provide legal reasoning. Simply using an arbitrary die roll would not be lawful, unless the legislature specified it.

Using a die roll would be arbitrary and capricious (and thus I assume illegal), but would it violate equal protection specifically? I suspect not.

It's probably illegal, agreed, but because it wouldn't pass even the rational basis test for a Second Amendment infringement. It definitely wouldn't violate the equal protection rights.

I don't believe the legislature has to specify the criteria used, if they explicitly give the local officials discretion, but their criteria do have to have some rational objective.