For the G&W-Bowser matchup, I feel that Usmash is one of our most important tools as it helps negate a large portion of his aerial game. The invincibility on that move is fantastic and should not be discounted so quickly. Bowser's Down B hurts, but it's also extremely laggy and predictable; unless he hits you with the grounded part of it as well, you're likely not going to have to worry about it all too often. If anything, the aerial tool to fear most would be his retreating bair, since even though we can technically block it it's a fairly safe tool for him to pull out.
On ground: Bowser is juggle city for us, we combo him like crazy if we can get in. Unfortunately, however, that's somewhat difficult for us as his ground range is fairly threatening. Crouch-dashing combined with one of the fastest dash-to-shield conversions in the game helps our approach somewhat in that we can avoid some of his more dangerous tools, but it only goes so far and it's still not perfect. Bowser's Side-B, while kinda bad for him as a stock conversion move ala Bowsercide, is still a command grab with respectable knockback, especially at higher percentages. If you get grabbed at around the mid- 80%s or so, you're likely going to die, so at least try to take him with you
Also, as has been mentioned, Bowser's OoS UpB is a pretty nasty tool to have to work around, effectively forcing G&W to have to play a lot safer. Of course, that's moreso only a problem if we're forced to approach.
In the air: Bowser has range, but he also has a decent bit of lag on his moves. Nair is one of his faster tools, but its range and knockback are moreso suited to being a "get-off-me" tool than an actual killing move. Our fair beats it fairly well, so it's not too much of an issue unless we get ourselves into a bad position. His dair, like ours, sends him at a specific trajectory, except that he actually has even less control of it. On top of that, its beginning animation provides an obvious clue that it's about to come out and is slow enough for us to easily react to in most instances. This should also largely be a non-issue. His fair has good knockback and speed, but its range outside of what's directly in front of him is somewhat lacking. I'm not entirely positive on this, but I believe our fair slightly beats his out range-wise. Definitely something to watch out for, but more in the sense that you can't just throw moves out above/beside him willy-nilly. Lastly, his bair... oh goodness, that bair. Perhaps I give it a bit too much credit, as it is still punishable if he randomly throws it out in the neutral, but the range and power on this move is phenomenal. and the end lag is hardly there to compensate. As I mentioned before, it's probably one of his scariest tools in the air, particularly if he RARs to fight our fair.
In terms of recovery: G&W > Bowser, easily. Bowser's UpB is great punish move and can still cover a good amount of range, but it's slow to reach its maximum distance and is easily punishable. If Bowser's offstage, you're in a good position; if he's below-stage, you're in a great position. His horizontal recovery is better than his vertical, so once you've got him off, grab the ledge and box him out. Bowser will also have a hard time gimping in return, as Trampoline Jump covers the majority of its distance pretty quickly and has lots of invulnerability.
Projectile-wise: Sausages trump Fire in terms of usability. Bowser really shouldn't be using fire in this matchup, as a) it's easily bucketable and b) there aren't really any situations in this matchup where fire is a better option than any of his other moves. Since Bowser also has a large frame, it's rather easy to put him on the defensive if you opt to use the foodstuffs for the mixup. It's not the best damage, but it's pretty free so eh.
In summation:
Pros:
+Ohhh, that upsmash
+Projectile superiority
+aerial superiority... for the most part
+ability to gimp, pretty difficult for Bowser to gimp in return
+Crouch dodges a good number of moves
+Pretty large target
+Juggling
Cons:
-Bowser's aerials can provide a decent challenge
-Bowser kills much earlier than we do; we're light, he's heavy
-Bowser has good range and decent overall speed, both lag-wise and movement-wise
-Bowser has a great OoS punish option
-Bowser has ground superiority... for the most part
-Bowser can keep us out pretty well
-We take quite a bit longer to kill him, and rage only helps him get better at killing
Neutral:
=We both have to respect one another's range. Neither of us completely dominates the other in that regard.
=We both have to be wary of the other's positions and options at any given time; every move has to count.
=Though each character technically wins one area of combat (G&W in the air and Bowser on the ground), the dominance on both ends is more 55:45 than anything. Even a slightly better player of either character can feasibly win both.
I hate to give such a trite response, but honestly I'd put this matchup at 50:50. If I were to lean either way, it'd be 55:45 Bowser, but unlike Ganon he doesn't have a fairly reliable way to guarantee the kill. Yes, he hits hard, but he has to hit with a viable killing option first, and quite honestly he's going to struggle with that. I didn't mention much about either side's tilts, though, so someone else might want to cover those in greater detail.