A flint eater?! While not as evil as a lock that just won't spark, they're not the ideal. And this is exactly why you will see me recommend a semi-custom when someone is considering a flintlock.
If you're inclined to tinker with the lock you may be able to correct things.
Try switching the bevel up or down.
Next look closely at the angle the flint is when it first contacts the frizzen. Is it making a straight-on collision? Or does it glance downward along the frizzen face?
Chances are, a flinteater strikes straight-on. Try using a shorter flint if you can find a supply. Try using a thinner piece of leather in the jaws. Or, I've gone to the extent of cutting a small hole in the leather to line up with the jaw-screw. This allows the flint to be set back a tad farther. The shorter the flint, the farther the cock will travel before the flint contacts the frizzen. This makes the flint strike at a better angle.
It's possible that the frizzen spring is too strong. That's the spring that puts tension on the frizzen and holds the frizzen in place. If this spring is stout, the collision between rock and frizzen can be mighty severe. I've weakened the spring by judicious filing. Careful! You might want to buy a back-up spring before attempting this (they shouldn't be too expensive). File with the lenght of the spring, not across it. Then polish the spring as best as you can. I'm told any scratches that go cross-ways can become flaws that will break the spring. Fortunately, T/C parts are available and not outrageously expensive - mistakes cost money, but so does education.
Yes, BP fouling will draw in moisture. But some BP is worse than others. I have no idea what the quality of Goex is nowadays, but Swiss is pretty dry. It's a funny thing tho, in the bore, that "dampness" is good for making many shots without swabbing possible. In the pan it's bad.
I prime with 2f. You can also use 3f. But 4f does not have a coating or it isn't polished. The coating that 2 and 3f have helps delay the moisture soaking in. If you're on the range, and you know your next shot will be right away, you can probably get away with priming with 2f or 3f. If the circumstances dictate - you may have to really wipe down the pan and frizzen between shots.
I had a misfire last deer season because of this. I used my normal 2F prime. I loaded and went on a deer drive in a moderate rain. The gun went off on my first shot at a deer (missed the deer tho). I reloaded, wiped the pan, and went on. At lunchtime I decided to light it off into a stump. The pan charge was soaked. Although I did wipe the pan and frizzen off, I didn't do it well enough and the fouling drew in moisture. Keep in mind this is with Swiss powder, 2F, and a Jim Chamber's lock. So even the best quality components aren't idiot proof. But I at least feel confident that with these components realiable wet-weather performance is realistice - not futile.