When I meet a LM I immediately see if they pass the for glory test - I stand at the edge of the stage, they come in with either a dash attack or grab which I shield or spot-dodge (getting thrown by LM is no big deal, he doesn't have any followups if you DI away and jump to avoid an up b if he dthrows, so shield if your spot-dodge timing isn't on point), I grab and back throw, they panic and use their second jump, I full jump and fair them twice. That's a stock.
If they pass the for glory test it gets harder. A good LM conserves his second jump, only using it from the relative safety of a low recovery. So the good news is, a LM of almost any skill level can be baited into an approach resulting in either Bowser getting thrown (no big deal), or the LM getting thrown off the stage and into a possible lost stock scenario. If he recovers low, good news again, Bowser's back air trades very favorably with LM's up b. If you run off the stage into a back air you will stage spike him if he doesn't tech. If he's countering your offstage fairs, that's him telling you to bait his counter and fair after it next time. The only thing I don't have a good contingency for is if LM uses his side b to snap to the ledge. Perhaps dtilt could intercept it maybe flame breath, though you'll be risking a bad trade. Using flame breath against an offstage LM doesn't really work if he goes low - it won't stop his up b. LM's side b will go over the flames onstage also so it's not great in the neutral. The point is you've gotta capitalize as much as possible every time LM is off stage. Besides hitting LM under the stage, don't go out of your way to bair LM, if he's recovering from above he'll be looking for a chance to throw his counter out. Bair into that and you might die. If he counters you'll be much safer having thrown a fair, and obviously even safer if you get uncomfortably close and bait out the counter before hitting him.
The problem with turtling too much is LM's charged smashes have too much knockback to punish on shield, and if you're close to the edge you'll be pushed off the stage. I don't recommend charging into them either. Sometimes you just have to take the hit to shield and get out of there. This isn't a match where you can throw out big moves and expect to be given enough time to shield. LM can punish almost anything from across the stage with a dash attack, he's that fast. So you'll need to stick to safer moves. Little Mac's often have a roll habit, and why shouldn't they... he has a great roll. Early in the matches, it seems like a good idea to fortress to catch that roll. Problem is, if they catch on and bait that fortress you'll be eating a smash every time. At that point it's not worth it, and fortress only gives you followups if the other player DI's in. Better to punish rolls with something else like pivot grab if possible. Maybe utilt too, but LM is small enough to hide inside Bowser's body and escape its hitbox so be warned.
When he get's his KO punch, hopefully he just got hit by something that killed him. If not, just don't throw out any big attacks. If he's about to get it, see if you can put him at disadvantage as he gets it so as to maximize your chances of hitting him before he hits you. If he's on the ledge, it's tempting to throw out a fortress. It'll cover a lot of options and take his KO away, but if you miss you'll be eating a KO punch so... consider jabs, consider pivot grab... good luck. Don't use flame breath.
More than most characters, Little Mac punishes aggression. If he's charging a smash you can be brave and try to grab him out of it, but generally you'll have to give LM respect in those moments. You really don't want to eat charged smashes all day. Canonically it should be the other way, but Bowser has to carefully circle the small elephant that is LM in this matchup and take any openings he can. LM is the aggressive one here, that's just how it is. Prepare to spend some time hanging on the ledge too, comes with the territory - LM will rarely let you land on the stage. His dash attack will punish any onstage landing strategy, and LM will find a Bowser bomb or dair trivial to dodge.
Little Mac can be very easy or very frustrating depending on the player's skill. Unfortunately I've played some really good Little Macs, and they can make it seem like an impossible matchup at times. But just remember, all you need is to get him of stage and correctly approach the edge guard to take a stock, so you can turn the tables pretty quickly if you catch LM at disadvantage.