A 30mm round plowing through someone’s living room because some bath water IQ nitwit thought they saw an alien is not a valid use of government resources. Even if they’re from Jersey.
You send in a supersonic fighter because no conventional drone is going to escape it. Dogfighting it isn't necessary, it's doubtful they'd detect you at all if your fighter is loitering at 10,000ft. Eventually the drone is going to run out of power, and you can keep sending more fighters to relieve whichever jet is on duty (if necessary).
Apaches are cool and all, but if cost is your concern then it's probably cheaper to send a single pilot in a single-seat F-16 even if the avgas costs more. Even if you gotta wait 4 hours for your target to go home, it's still probably cheaper than a single AMRAAM.
What it would come down to is that an Air National Guard unit is well prepared to intercept something in the sky (although a slow, low flying drone might be tricky). It's pretty extreme to actually do that, of course, and just observing the thing would always be an option.
An Army National Guard unit might have Apaches available, but putting one in the air in short order to perform air intercepts is not their mission.
Drones are generally too slow and have too small of a radar footprint to engage with a fighter jet. Helicopters are a better tool. Unless they're very large drones than most missiles on the jet won't really be applicable. You can't shoot down a tiny front with radar or heat seeking missiles.
Let's not forget it took how many sidewinders to take down the Chinese balloon? More than 1 makes someone look foolish.