If that style is giving you a lot of trouble as fox that probably means
1) he has more techskill than you,
2) there's a fair amount of idle time where you aren't really doing anything (taking stage, retreating, comboing)
3) you don't understand when you can get out of things falco does, and how to do things falco can't get out of
At least one of those is probably true.
Pillars only works while shine doesn't knock down. Regardless, after a certain number of shines you get sent into this weird no tumble floating knockback state anyway where you can air control in two dimensions(ever get shined and randomly float around? you're actually controlling your location with the stick!) you can get out once that happens. Just mack sure you di down and away and don't tech predictably!
Remember falco is awesome at verical mobility and fox is awesome at horizontal mobility. If he likes to laser on the ground a lot, take to the platforms and drop down on him. If that doesn't seem to help much initially, practice wavelanding a lot. Platform mobility is huge as fox.
If he's dash attacking you, keep space, try to bait it out. This is hard if he's also got an itchy trigger finger, because you can't really dash dance that much. Note that falco's grab isn't that good and shield makes punishing dash attack basically free. You can just wavedash out of shield and grab or usmash.
If he's dash grabbing a lot, just run away, or hit him with a move! XD If you ran away, just don't run too far so you can turn around and hit him back. If you want to just hit him, I suggest utilt (sets up combos), nair(not as punishable as utilt), or dash forward jump hold back on the control stick then dair (should look like you're slightly retreating or just dairing in place after the dash).
Remember as a mixup to drill shine you can drill grab because you've got a chaingrab on falco from 0-66% with good di and 0-86% with bad di.
Try hitting with upairs after uthrow too! Uair combos into uairs unless he's got great di.
If you nair after falco bairs, you're hitbox will beat it out since dair leaves his whole upper torso exposed.
FH over lasers is better than staying in shield but it's punishable if he gets a read on it.