It seems like some of you birds are showing too much respect/fear of MK's air game.
If a competent MK is beginning to use short hop fairs, bairs, or nairs on you, he's showing you no respect. The range on those are quite bad, and the ending lag is bad too. Short hop fair, bair, and nair all do not auto cancel and are very punishable upon landing. I think Falco can stuff all of these options out with good ftilts or RAR bairs, or simply wait and punish them.
MK can do the Brawl-esque d-air camp on Falco a lot better than on most of the cast because Falco doesn't have broken OoS options or overpowered range/disjoints/tilts. However, the horizontal reach on it is quite awful and should be quite easy to see coming. At the very worst, you should shield them and he'll jump away (actually, he could feint a d-air if you're too conditioned and just fast fall -> grab you aka tomahawk). If he's being cheeky with the short hop d-air/aerial d-air camping, I think a good tilt or bair will beat that too. Aerial MK is much easier to bair than a grounded MK.
Often, the way a high-tier character with a good neutral beats our bread n' butter dash attack/dash grab approaches is either by walling us off with aerials (C Falcon bairs, Sheik fairs), jabs (nonsense jabs like Lil Mac, Captain Falcon, ZSS), or just being better than us at our own game (C Falcon dash attack/dash grab lol on top of the rest of the aforementioned). Actually, Meta Knight is at his peak when his enemy approaches him and he's able to bait and punish their mistakes. Falco is similar because Falco's own approaches are meh... but his ability to force approaches is alright against most of the cast because LASERS (despite how nerfed they are). I don't know how Falco's jabs interact with MK's dash attack, but I'd imagine that to be a huge part of the MU. Otherwise, it honestly seems like dtilt, ftilt, RAR bairs (really precisely timed to hit him grounded, can be sloppy against aerial MK) will be your BnB here, because I feel like that's just Falco's general walling off options against most of the cast (+ jabs and the seldom reflector for short hop approaches). Falco dtilt should clank or outrange any of MK's stuff except for aerials (and remember, MK short hop approaches are bad!), ftilt should outrange anything from MK and seems to be superbly safe when spaced properly and not whiffed or power shielded, and RAR bair should beat his aerials and even grounded if he's greedy.
The really sucky thing is that one of Falco's coolest and most powerful assets, his off-stage game, is ridiculously risky in this MU because... Meta Knight. If you go really deep to edge guard Meta Knight and he air dodges you... expect a counter b-air or d-air from him because lol 5 mid-air jumps. He will almost certainly never get hit by your d-air (almost no one does, curse Falco d-air nerfs...), a quick drop -> bair may catch some MKs off-guard if they're recovering low, fair seems too telegraphed to hit any MKs imo. You can't really scare him when he has so many jumps and very extreme recovery means via shuttle loop or drill. Oh, and be aware that Drill can STAGE SPIKE you! It spikes you towards whatever angle he is directing it. But if he does a very obvious one, that may be your cue to actually use dair lol.
MK edgeguarding against Falco goes like this. If you keep using phantasm to go on-stage, he'll just prime Tornado or, if you're REALLY too predictable, spaced F-Smash it. If you use it to grab the ledge, that should be safe most of the time. If you're too obvious, he may just Tornado the edge of the stage instead. So phantasm recovering becomes this weird 50/50 thing sometimes if MK is in range to tornado either the ledge or the stage instead of just the stage. If you're recovering low and do an obvious Fire Falco (e.g. you foolishly wasted your double jump and are still low for whatever reason), he can melee shine spike you with his d-air on your windup and possibly even during flight lol. Or meet you with a bair, which is also really bad to eat while recovering. If you've grabbed ledge, his optimal punishes are to use shuttle loop if you ledge jump, otherwise a tornado, spaced f-smash, or grab (or lol dash attack because MK) to meet your roll/getup/attack options. But that all happens really fast, don't be predictable and it shouldn't be too bad. He can YOLO Drill you too on last-stock or unfair stock trade situations with either the spike or
knockback similar to Ganon side-b suicide.
Sorry for theorycrafting so much, I main MK and play Falco quite a bit but I've never Falco vs MK and very rarely play MK vs Falco. The only Falco who has beat me soundly only played once before leaving. He utilized Perfect Pivot Jabs very often and extremely elegantly, which is why I felt that his Jab is a decent-good option against MK's approaches. When I play Falco myself, I feel his dtilt and ftilt are great stuffing and enemy-halting options, and RAR bair is such a good option against characters with GOOD aerial approaches like Jigglypuff, so I see absolutely no reason you can't contest MK's cruddy air-to-ground game with it as well. Force MK to approach you through whatever means. MK making you approach is what he wants you to do, and even a top tier approaching MK isn't fun for the top tier (well, except for Diddy). Falco himself doesn't have the greatest approaching tools :\.