Many avoid due to fees and attitude. Im close, but avoid…$40 in parking and car use fees turns me off. Too many other spots close by minus the negative stuff to visit.
KCRG (Craig) is about a 40 minute drive from Fernandina/Amelia Island. 2 friendly FBOs, rental cars (in normal times anyway). Uber from there to Amelia might be reasonable. Fuel prices at KCRG reasonable, but plenty of airports around with better prices.
There's also a public grass strip airport about the same distance away (Hilliard Airpark). I'm not very familiar with what services they might have there, but according to AirNav, fuel prices look pretty good. (https://www.airnav.com/airport/01J)
If you're looking for nice beach areas in NE FL/SE GA also take a look at KSSI, Jekyll Island, and St. Augustine. As Butch said, lots of options.
IMHO if you are already going to rent a car while here, fly into a different airport and save the high fees at KFHB. That will go a long way towards paying for the car rental.