Fox's FJ basically does him no favours vs grounded Bowser because of how the outcomes of Fox's falling aerials vs Bowser's shield and Up+B (and the rest of his character) works out.
Bowser also fares decently vs styles that emphasize intensive platform play (the bair footsie from side plat) and is generally good at combat at high altitude. He's good at these because of his enormous aerial hitboxes, his strong air grab, and weight. The air grab forces opponents to exit shield or they get *****. This usually means jump, which is where enormous hitboxes is useful. He can get nicked by a lot of weak hits and counterattack with some manner of Up+B or FF to land first and hit them out of their attack's duration (especially good vs sex kicks) and so forth.
This is basically why Bowser, though bad, has a semblance of a cohesive game vs Fox. Counterattack characters tend to do well vs low range characters. For Fox to win this like the unfair match it is, you have to tug Bowser along with your mobility and abuse that Bowser has almost no stage presence outside of his immediate area. He has terrible reach and commits heavily if he moves forward doing anything. Grab him and shark his descent. Emphasize hitting up tilts, uairs, and strong sex kicks for comboing or it probably won't work. U-smash to uair works if they DI the u-smash in, so use the edge trap..