I think there are some retroactive reasons for why Bashir gloats a bit about significant medical events like reviving that dead guy in the beginning of "The Passenger"
(1) It really is a pretty impressive thing, and genetically enhanced or not, Bashir is young and wants to get some credit. Maybe a least part of him really wants the acknowledgment of his great deeds.
(2) Maybe Bashir thinks that he can actually distract people away from the great dead (which might lead people to question his abilities) by gloating a little too much and therefore distracting people by having them think he's a annoying stuck-up know-it-all
Either way, it relies on Bashir being a very good judge of the people he talks with.
Can you just imagine what his life is like, constantly having to lie to people and judge how they react to those lies?
I always thought it was weird in the early seasons how he was a bit of a ladies-man with women (like at the beginning of "Q-Less"), but everyone else thought he was a stuck-up kid they didn't want to hang out with, and he was always so clueless when it came to dealing with Garak.
I think Bashir, even early on, is much more capable of being tactful in situations and controlling conversations they way he wants them to go. He would need to have honed those skills a little to keep his secret. But he almost purposely acts up a little too much (too annoying, or too proud, or too whimpy) to keep people off-kelter.
In early seasons the only people he ever uses his full-blown conversation skills on are the women he dated. He is a young man after all, into sex as much as any other (as far as we get on TV).
But I would/could also argue that Bashir is really still trying to find a balance. Still trying to know how much to lie and how much to reveal his true self. Remember, he didn't even know about his own genetic enhancement until he was 15 (stated in "Doctor Bashir, I Presume"). So his genetic enhancements couldn't really be all that much beyond regular people, or he would/could have been found out earlier. Maybe in these early seasons he really is what he seems, a brilliant young man trying to find a balance between pride in his work and modesty, while only occasionally worrying about getting "found out".