Basically it's a war of attrition, but we can win it just fine. You can dash forward, shield, see if you bait his rolls. You can catch his roll towards you with a pivot f-tilt or just Up-B. Punish roll-away or spot dodges with Fire.
Best time to kill him is off stage. Bowser can go pretty far out/down and still make it back if you keep his second jump. Bait his Up-B with an empty jump, because they'll almost always use it to escape your hit. Catch him on the ground after he uses it. If they keep their double jump, you can catch the double jump with your own double jump into Up-Air.
Just keep pressuring him off stage and catching him with he lands. If he tries to recover low with Up-B, just RAR off-stage and stage-spike him with Fair. It's easier to hit with than Bair. If he recovers high and you bait out his Up-B with empty jump, just follow his air movement from the ground and try to catch his fall with Up-Smash.