Dcold. Sheik is not at the overall advantage, seriously. Bowser's only real fear is the FTilt lock, which works on Bowser just like every other character, BUT landing it on good Bowser's is not a straightforward task all things considered; that's not to say the FTilt lock is the only thing that works on Bowser, he is very capable of being combo'd and juggled if he is not careful. Good Bowser's though, are well aware of the threat the FTilt lock poses, so playing extra cautious to keep Sheik away is only natural for Bowser's during this matchup. Bowser's will keep a reasonable distance and deal with Sheik from there and don't think Bowser's can't hurt you when he is not at grab range: Klaw hopped aerials (preferably Fair's) and specials (Flame Breath is most common), spaced Fair's, grounded Klaw's (unlikely), DTilt, directed FTilt's, Flame Breath, these are all more than enough to keep Sheik at bay and she really doesn't have any rock solid methods of getting in there, well timed SH aerials or a dash attack could work, but they're risky and if shielded/powershielded she is prone to getting grabbed or Fortressed, both being very effective methods of dealing damage, knocking back and getting breathing space. These moves all deal hellish amounts of damage and have frightening KO potential on some of the lighter characters, from about 90% onwards Sheik is at risk of being KO'd by many of these attacks, you cannot sponge them for extended periods of time.
I'm not saying Sheik is at some huge disadvantage, she has the ability to combo the hell out of Bowser if she can catch him off guard and punish mistakes efficiently, but breaking through the castle that is Bowser is no simple feat, especially considering that Bowser does have an answer to Sheik's common options while Sheik has no surefire method of piercing Bowser's defenses and getting the combo's going. Also, don't underestimate the Fortress, it is a deadly attack and will make you think twice before jumping right into Bowser like some bold fool. KO-ing Bowser is tough because of his weight, gimping his recovery is an option but don't be fooled, it's not something that'll happen every stock; the recovery, though predictable as it may be, can cut through most aerials if the right hitboxes connect, if you want to easily gimp it you'll have to hit the sides or underneath and Bowser's like to cover that weakness as best as they can, of course this does make edgeguarding a brighter possibility, though that does depend a little on the stage.
40:60 in Bowser's favor is a realistic conclusion, but 45:55 is almost just as reasonable; in fact you know what, the former is probably too harsh, but the latter may be a bit too inaccurate, it's still closer to 40:60 though. Dcold, this doesn't mean Bowser is a counterpick, just that the matchup can be tough if you don't have enough knowledge or don't know what you're doing.
@ less than 3. Ok no, You have written off Bowser's entire metagame. If you seriously believe Bowser has no approach options on Sheik you are sorely mistaken.
First I'll address your claim that Bowser has no viable approaches. Are you unaware of Bowser's entire movepool? Please read my entire post again, it's not a rare scene that a Bowser will powershield his way through needles and get within jab range, then there's Fair and Flame Breath and Fortress. You may not believe it but these are all simple, viable methods of reaching and damaging. You cannot just say "get Bowser in the air and its all hunky dory", Keeping him in the air isn't as simple as you might think, and don't for a second think that his DownB is the only means of reaching the floor because it's totally unsafe and good Bowser's wont do the predictable.
You can tell Sheik's to stay out of grab range but that's a lot easier said than done, Sheik cannot cause problems for Bowser without being dangerously close and at risk of getting grabbed and Fortressed. Speed is of little issue here, Bowser has one of the best jabs in the game and one jab is all it takes for a setup. Sheik's shouldn't be aggressive, they should be cautious, obviously, but if you think Bowser has no way of approaching Sheik but in a reversed situation it's the complete opposite then you clearly know nothing about Bowser. Don't act like the needles completely ruin Bowser, powershielding works fine and so does jumping over it.Also, explain how Bowser is so crazy easy to gimp, and exactly how Sheik is supposed to get him into such a position without taking serious risks and damage herself. You don't seem to know much about this matchup and appear to be just theorizing. Bowser is a hell of a lot more versatile than you say; You're simply underestimating his abilities which is just as well because that's what everybody else does.