I'm assuming the fairies can't just make a fake dragon egg and give it to the king.
One can theoretically see around 208 miles from the top of Mount Everest. If we ignore the added height flying fairies would get, this is the earliest warning they would have of an invasion. We can cut that distance in half so the fairies can be absolutely certain it's an army (and also to give a fairy time to look in the right direction). This gives them some time to plan and prepare.
Unfortunately, given their isolation, the fairies probably don't know what the army is for. You don't give a flight speed for your fairies, but going off the lifting data each fairy can fly (carrying 1/100 of a tennis net) about 4.5 miles in a 12-hour period before requiring rest. If we assume fairies don't like to camp out, then the army will be a little over 2 miles away from Mt. Everest by the time they're close enough for a fairy to visit them and fly back without rest.
Assuming the humans and fairies speak a common language, it seems like a fairy would quickly discover why the humans had come, probably by being questioned at the point of a sword about "the dragon egg of immortality". Since he doesn't know, relations would ... quickly sour. Luckily, due to his relatively small size and flying skill, dodging arrows and the like should be pretty easy. There's a potential subplot in here about rescuing other fairies who set out when the army was 4.5 miles away and were captured while resting, which could be used even if the fairies don't end up fleeing, but I digress.
It's surprisingly difficult to find a source on the weight of tennis nets, but this listing says one net weighs 20 pounds (there's also a 'championship' version that weighs 22 pounds, but I'll stick to 20 for the purposes of round numbers and skewing numbers against the fairies). If 100 fairies can easily carry 20 pounds, that gives the entire population of 50,000 fairies a total lifting capability of 10,000 pounds, or 5 tons.
Tennis balls are around 2 inches in diameter (and I'm assuming fairies can fold up their wings, or have architecture that's flatter and wider than human housing to leave space for the wings). In any case, they would comfortably fit into a 1:24 scale dollhouse. Again, it's difficult to find a weight for a dollhouse, but most sources seem to indicate a full-sized house at this scale is around 10 pounds. These are the ones meant for serious collectors, so they include furniture and presumably are made out of sturdy wood (I was also able to find dollhouses that were 2 pounds, but those were flimsy).
If we assume 4 fairies can live comfortably in a single dollhouse equivalent, that's 12,500 fairy households. That's a total weight of 125,000 pounds, or 12 and a half trips for the fairies. They can relocate to pretty much any mountain peak, but the closer they settle the more time they'll have to carry things from their old home. Let's assume they choose a spot 12 hour's flight away. This would take them 14 days to do, including rest and the flight back. On the last trip, since only half the fairies are needed, the other fairies could perform some of the other actions suggested here to slow the advance of the army if necessary. In fact, it should be possible to split the difference between all required trips so that a group of fairies is constantly harassing the army. Obviously, this should be done such that nobody has any idea it isn't just bad luck, since the king wouldn't take kindly to sabotage.
Assuming the army takes 20 days to reach the summit (a gross exaggeration), the fairies still have enough time to make 5 more trips, carrying an additional 25 tons of assorted food, medicine, seeds, livestock, and/or pets to their new home, where they can wait for the king to get tired of searching around in the oxygen-low, freezing cold mountain for a dragon egg that doesn't exist.
And in all likelihood, the fairies don't have nearly that much stuff, their new home would be much closer, and the army would take way longer to scale the mountain.