I'll add some personal info and things I know happen in the MU, so this isn't just me vs Arty and such or something else.
Kind of a bump but I think it's better for this one since I know this MU kinda well.
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Mindset of don't approach is the correct one. Make sure you get down silent lasers so Lucario can't crouch or crawl under that and has to play shield games with you.
Laser is defiantly good, duh, a lot of it though is learning the spacing of what you do next though. At certain spacings, if you try to side B through Lucario as a get away, he could roll backwards to avoid the activate hit and punish at the other side. Same for a couple area where he can Utilt and Dtilt Falco out of Phantasm. It's just learning the spacing of where and when Side B is safe to use. Lucario is a bit better off with the CG since it really isn't too bad compared to other characters. It shouldn't work kinda, but for now 3 Dthrow to Dair is fine at 0%.
Jab is also a good move to use. Same reason Ike is kinda harder than most characters around him a frame 2 move with that range outspeeds Lucario on the ground for all his option. This is good but there is something to note.
Depending on how you follow up after Jab 1 or even Jab 2, Lucario can Force Palm, side B, you out of it.
Since Force Palm is considered a special move if he does it within 8 frames of landing he can initiate the move on frame 1. So if you try to do Jab1 > grab, he can Force Palm out of it. Just learn to mix up how you follow this up so he can't do it off every jab and make him play guessing games since Force Palms after lag is long. Just keep this in mind when your jabbing him. Don't do Jab 3 ever he'll just SDI up and footstool + dair.
Lucario most used move is going to be fair, it will be. All of his cross ups will usually start up with this move. Falco's bair will eat threw that every time if he commits to it too much, still be careful since Lucario is better in the air and at juggling that vice verse. So with fair if he hits your shield while rising he has these options to mix up, that are viable. Uair, Fair, Bair, Dair, Nair, Force Palm, Grab, Aura Sphere, Jab, footstool. Depending on where he lands and how fast he can turn depends on what he does.
All the aerials are pretty easy to see as self explanatory, even the footstool. The grabs are kinda easy to see as well. Same for the jab comment I said before this applies here as well, Lucario might try to fair behind you to Force Palm or grab on landing, if he does grab correctly he will turn around immediately when he lands to do it. This is to cross up on shields. If you know he is trying to do this, Bair OoS. It will punish him everytime.
Aura Sphere isn't as easy to stuff out as it looks like on paper. While it is Lucario's best move, keep in mind having a reflector isn't going to solve it necessarily as easy as Fox or Wolf. Kicking it gives more surprise, it's not as easy to predict the reflect and even then to react to it when it's not always static with spacing. The down side is the lack of quick cooldown, which is what charging Aura Sphere might be a baiting tool for. Lucario might be baiting the reflector, so try and wait for him to use it if you can. The main times he will use it is on a read/mix-up or more often than not when your landing or recovering. It's easier for him to punish this way and avoid the problem that is your reflector. Otherwise just having the option is goo enough for Falco since it makes Lucario have to be wary of it and he can't abuse it on someone like Marth/Snake etc whom he can use without as much worry of retailiation.
Fsmash is something a lot of people whine about, it's not that good when you learn how to fight against it properly. If you learn the timing of it on the ground, it takes 21 frames to start -up even with the amazing cool-down thanks to Aura an other things, if you Power Shield it, it's really moot. When your grounded and he just throws it out there like that, just Power Shield then Grab/Dacus/Fsmash, etc. He should be trying it when you land, just be good with covering landing and recovering to avoid this hitting you.
Kinda related but while Falco has a great spot dodge and air dodge in a way, Lucario's hitboxes linger, that tend to make both those options not as great. Shield is better against him since he has more problems with that than someone using a spot dodge or air dodge. Don't go for juggles unless you are going to punish dair, Lucario's dair will stuff anything you do vertically, so go horizontally aka when he does run back then bair him. If your being juggled remember to DI and SDI away and not into him. Because if you DI into him, your giving him more string an combo potential. Going to the reverse, if Falco is trying to catch his landing, be mindful he can wave bounce with Aura Sphere and Air Dodge whenever he wants. So be patient and try and read where he is going to land, just don't stand right under him, but close to him.
Lucario has an amazing roll of course, but rolling into people is still bad if they read it. If he rolls backwards, just keep shooting lasers and make him come to you. If you think he is going to try and roll into you, turn around and Fsmash, Usmash, you can and will catch him with Falco if you react fast enough. If you get Lucario off stage, you can use Bair on him to edge guard and be careful or Aura Sphere/him turning himself around to use bair with a wavebounce. If he gets too far away to get back on without his UpB, hang on the ledge and wait, if he goes for the ledge thinking you will get off, he will die on most stages. If he tries to go right over you at stage height, jump and Dair spike him. If he goes high, punish his landing. If he goes at an appropriate height where is is safe from the approve, get on stage an Dacus or anything to punish him.
Note he can wallcling on certain stages even without his UpB, Yoshi's, FD, Frigate, PS1, etc. Remember where he can and if you know he is going for it, just Dair him/Footstool him. He will die after wards since he can't UpB again.
Aura, yeah it's gay and why this MU is easier for him. Falco has trouble killing a lot in this MU, be patient and don't force it. His damage per hit is strong but you have safer outlets to damage. Keep using these and avoid the situations where he can gain a huge hit. It can be frustrating at the end, but just don't force the situation and you'll be fine. Know what he can do and remember what you are threatened by in this situations.
Stages, Game 1, Don't let him have YI or Lylat, he does better on both those stages. Aim for BF or SV.
Game 2, take him to one of the above + Halberd, Halberd is Lucario's worst stage if not one of his worst.
FD isn't the best CP for Falco, Lucario actually gains a bit from it in this MU where platforms like BF are more preferable for helping Falco land and stop juggles, FD lets him live longer and helps him get more chances to punish landing and recoveries. I really think any of the above 3 are better.
Ban Frigate, it's the worst of anything he can take you to, just take my advice it's the worst stage for Falco in this MU.
In the end, I think this MU is even, 0 on the BBR chart, maybe a smidgen either way not sure but its even based on top level results and on paper theory. Lucario isn't super common but it's a MU that Falco can handle fine if he knows what he needs to do.
Just a lot of random info I think I know and personal experience but playing as Falco and Lucario, I've gone to Falco as a secondary sometimes.
I hope this helps a bit.