Is it possible to get a permit to fly non commercial drones within the borders of this forest?

Yes you can fly here if you notify the two ATCs nearby

No!! Are you high!!

I've flown my drone within this forest before. The GPS recognizes that the forest is a "Fly With Caution" zone. You do have to accept full responsibility for your flight, but it's not a No Fly Zone. The FAA does require drone operators to register their aircraft on their website. It's only $5.00 for sUAS 107 registration, and it's good for 3 years.

Not sure about the legality but people fly drones there all the time