3 things you really need:
1.) It's easy. Tricks by the ledge.
2.) Juggle combos (u-throw -> u-smash).
3.) A method of getting around lasers (try jumping over, sounds simple but works, or u-smash out of shield).
Fox played right can have an advantage on all stages. Just remember what makes this match different (lasers/shine combos) and predict it (mainly lasers) and **** accordingly. You should have a planned way around lasers. Why go into a match unprepared? Plus you're playing against an easy player for Fox to combo and gimp, so do that!
And seriously, you know that Falco's are going to double jump phantasm when they recover, so everytime Falco recovers jump wayyy out their and just follow him down into a shine spike. Don't try to be overly safe and stay staged against Falco. You need to get those easy shine spikes on Falco's predictable recovery. That can really change the tide of a match even if it's just mentally. Obviously smart (actually just decent) players will mix it up, but recovering by double jumping -> phantasm at the stage is just too good as Falco, because recovering low will end up in you being dead as well. So what you need to do as Fox is jump out where you think they will be to recover, if they just keep going low or whatever, shine anyway so that you can turn around, and then double jump back to the edge and grab it to get those invincibility frames. Which you can then use to **** Falco's low recovery. And of course watch for smart Falcos to just recovery really really high and not even mess with the edge. In this case, sometimes it's smarter to hit them in the opposite direction, because these Falcos will be assuming you're going to get the easy hit on them, so they're already DIing.