I play both falcon and fox so I pretty much know how to handle situations such as this one.
If you don't already know, falcon has a very "in your face" style of fighting. Most good falcon players will fight similarly so just use fox to his full potential and you won't have a problem. By full potentional, I mean LOTS of lasers. SHDL (Short Hop Double Laser), SHPL (Short Hop Pivot Laser), or simply SHL (Short Hop laser)*
a short explaination
Short Hop Double Laser - Short hopping and shooting two lasers before landing
Short Hop Pivot Laser - As you are running away from your opponent, pivot, Short hop, and shoot a laser
Short Hop Laser - Simply Short Hop straight up or while approaching your opponent (recommended), and shoot a laser. If you time it correctly, you will be able to hit CF in his torso area, which will also allow you to hit smaller characters such as fox or even ness.
If you can master these techniques and use them in a typhoon of laser tricks, you will be able to dance circles around CF. And remember, always be on the move so it's a lot harder to grab you. (but don't be careless and jump in above CF either...lol upsmash to the face)
Typically fox has a weakness to CF because fox is a character that is a perfect weight for z2d** combos. However I have found if you use these laser tricksies, it is quite easy to turn the tables against CF. Just practice these techniques and work on your mind games. Hope this helped!
*These are pretty much the types of laser shennanigans you can pull while short hopping. obviously off normal jumps you can shoot a lot more lasers. But those aren't important right now
** Zero percent to Death