Unless you're willing to admit that your character might be actually horrible, it's important to stop looking at "which characters are good against [your] character's bad MUs". A good portion of characters have a bad MU spread that's not above 50% positive, Bowser being one of them IIRC. That's not to say you should feel discouraged at all. Characters like Palutena have put in some serious work before, and by a lot of people's standards they're worse than Bowser. Every character has a shot IMO.
That said, what you want to be looking at more in a secondary is the MUs you personally have a bad time with. People might think, say, ZSS vs Bowser is in ZSS' favor, but if you do well against ZSS for whatever reason, it might be good to ignore that MU when taking a secondary into consideration.
The second thing is - what characters do you find fun? It's pretty self explanatory, but don't pick up a top-tier just because they're top-tier. It might actually be enough to pick a high or a mid-tier who compliments Bowser's playstyle if you enjoy them and can still do well. No MU is "unwinnable" and while some will be a bigger uphill struggle, having another character (or maybe two) to fall back on can be a saving grace.
Using myself as an example, I main Jr., who generally isn't that good at close range, can struggle incredibly badly with wallers, and sometimes has a hard time confirming KOs. However, he is kind of safe and campy and can take a beating while still having high mobility. Contrast that with my (current and hopefully permanent lol) secondary - Meta Knight - and you have an aggressive, kind of floaty combo and punish oriented game with a lot of KO power when performed correctly. I also think my Meta Knight also punishes my Jr's personally bad MUs like Rosa, Megaman, and Mewtwo, while still having a respectable place on the tier list (Which, while not a huge factor, having two low-tiers as your main/secondary could be risky since general character power can mean a lot, so looking up the list never hurt.).
Ultimately, even if the character you end up picking isn't a great contrast to Bowser (Say, DK or ROB, who are both heavy grapplers like Bowser. -though generally speaking a good secondary has some contrast.), they still might work, so don't feel disheartened if you find your main and your secondary share a common weakness. They both have a chance to win that MU and being able to mix it up and take the MU from both angles might be all you need to win anyway. I don't think any two characters can comfortably cover a whole roster without player skill being involved anyway.
I know this isn't traditionally helpful advice, but it's honestly the best advice I can give when it comes to choosing a secondary without spending forever telling you why you should or even need to learn how to play a certain high or top tier. To tell you who to play and who will win you Bowser's bad MUs is disingenuous, since the game is actually pretty balanced and it might not translate well in practice since it's more about the players in that situation. But giving you a gist of what to look for in a secondary seems like a good step in the right direction.
I hope this is helpful in someway.
TL;DR: In your secondary, look for comfortable and contrasting, no two characters can win you every MU without some skill behind them anyway, worst case is that you'll have two perspectives on a bad MU that you can learn to cope with much faster by broadening your options.