Check the regs re: firearms inside national parks. Far different than NF. They're not pretty and every year you hear of someone donating theirs. That being said, the juice ain't worth the squeeze on a short trip. You won't find anything to shoot on public land that isn't in or too near the park without losing most or all of a day.
I will nearly always have a personal defense firearm with me, but again, you won't be carrying it and it's best if no one even knows you have it. All the regulations still apply.