As a Lucario/Bowser player, these are my thoughts:
bowser's meaty limbs easily beat out everything lucario has, though this is the case for almost every matchup in the game from lucario's perspective. Bowser has high reward and great meaty, but he's slow enough and has enough end lag on most things that lucario isn't completely shut out like in certain matchups and has a fairly easy time grabbing bowser. Most of Lucario's stuff comes off grabs which helps in this matchup because bowser's OOS is really good but his dodges are ass and he'll have to pick some risky options (dodging or meatying) to avoid the grabs since lucario out-maneuvers him. A single grab for lucario = tons and tons of damage at almost any % and in general bowser is going to struggle a lot to land, and will usually be forced to the ledge. Things don't get much better there either as the reverse aurasphere charge option select is insanely potent against bowser.
For those who don't know, when an opponent is at the ledge, Lucario can charge aurasphere facing away from the ledge. if the opponent gets up in place, get up attacks, or ledge jumps, they'll get caught in aurasphere charge which combos into upsmash or dair or a grab. If the opponent uses get up roll Lucario just fires the sphere he's charging and hits them. The only other option is to ledge drop and aerial, but if lucario can react to this he can shield the aerial then punish with aurasphere when they regrab the ledge since they won't get invuln again.
Aurasphere charge approaches are great against bowser's massive frame and his only good way to deal with it from optimal angles is upsmash which is very risky. On top of that Aurasphere itself is pretty good in general as bowser's initial dash isn't super fast and his walk is super slow, meaning auraspheres at neutral are fairly safe because dash powershielding them is unreliable, though meaty ftilt will easily beat it out at almost any% so don't expect to camp him with it or anything.
I've been known for being really down on lucario as a character because I feel he's too weak at neutral to compete with fast characters with reflectors. Bowser has neither of these traits and Lucario as such is just fine at neutral against him, however it's key for lucario to stay very mobile and very tricky to try to bait out bowser's stuff. If you just run in like an idiot he'll just punch your head off.
Despite the obvious field day lucario has with bowser there's still the elephant in the room of bowser being able to kill lucario super early reliably, which is really what saves him in the matchup. He also has the ability to kill ultra early with moves like fsmash and can use the fear of these options to force lucario into picking options that give bowser free damage. In addition, bowser's bair is fantastic at intercepting lucario's recovery if you know the angle, but bowser has to be careful or he could get punished hard if lucario bends it around him then punishes bowser really hard while he waits forever to reverse grab the ledge.
Lucario Pros:
Projectile- forces bowser's approach
many viable mixup options
mobility
combos for days
juggles for days
ledge option select for days
superior offstage/recovery in the matchup compared to bowser
excellent grab game and ASC help mitigate bowser's crazy good OOS
Bowser Pros:
Kills Early with pretty much any move
hella damage at any%
outspaces lucario easily
won't die ever without aura's influence
Matchup Keys for Lucario:
Stay mobile and patient
don't air dodge into bowser and die at a super low% to something
abuse bowser in the air and on the ledge
Matchup Keys for Bowser:
quick reactions on normal moves to swat lucario, do not respect anything.
Know when your murder moves will kill lucario super early and leverage it against him.
punish optimally, being able to end stocks ASAP is EVERYTHING
Overall I'm confident that this matchup is +/- 0 with a very slight edge to bowser but not enough to make it +1. Lucario has the versatility to easily work around bowser's sluggish neutral game and forces the approach but bowser has the ability to swat everything if he stays a step ahead. The threat of lucario's grabs heavily discourages bowser from sitting in shield which makes the great OOS less relevant. On top of all that, Lucario can gain obnoxious advantages when bowser gets put in the air. However, Bowser hits like a train and will basically never have to worry about primal rage Lucario tearing him apart and killing him early, which in turn helps bowser get rage kills at pretty stupid %. Overall it plays out pretty evenly imo but can (and often will) snowball really heavily towards either side depending on who makes the right reads.