I got a lot better at this matchup actually.
If you aren't using Rest vs a skilled Falco, your battle is uphill.
uthrow > rest is unbelievably useful, so go for grabs whenever possible. If you attack with aerials regularly they will often begin shielding -- this is where you mix it up and grab. You can also shieldgrab their sloppy dair > shine or you can wd backwards from his attacks and grab him there too.
If you ever shield Falco's fsmash or dash attack, jump out of shield and rest him, or just grab him if he's out of range.
Utilt/uair to rest work but you'll be hard pressed to land them. I like to land falling uair's behind him then l-cancel into shielding.
When they become afraid of your grab and stop shielding and lasering, you can play with a pretty basic wd > fsmash game.
Lasers are what make this matchup a losing battle -- so deal with it. If you can keep up with Falco but get a Rest in, that's generally enough to win. If crouching will influence them into coming towards you (never for me), its probably a good idea. You can actually CC dair and grab them often too. I usually just stay in the air, watch out for random aerials, and bother Falco with fair/bair. When using fair, watch out for Falco's crouch canceling.
Chozen's Falco edgeguarding works very well for the most part -- hang on the ledge, and if they illusion on top of the stage, ledgehop and rest him (if he's too far away, grab him and bthrow him instead, and repeat)