the ninjak
FRINGE Division
If Batman is aware that Holmes is on the case then Holmes is screwed!
Even just using Batman Begins feats. Batman easily stalks and captures Holmes, takes him back to the Batcave and persuades him that they are similar in more ways than one. Especially considering Holmes is a recluse.
In regards to H2H. Nolanverse Batman is shown on film as being a heavy tactical brawler as I guess a homage to the the more classic Batmans we know from the past. (animated series Batman)
He isn't some Jet Li style fast paced fighter performing Kung Fu extravagant moves.
In Batman Begins Ra's Al Ghul/Ducard tells Bruce that "this isn't a dance".
Holmes seems more impressive than Wayne in battle. Yet Bruce hits heavily as proven by him shattering a wall in Dark Knight Rises.
If the fight becomes limited to Holmes vs Wayne in some room.
It would be a hard fight to assume who wins.
Bruce hits brutally and aggressively against men not familiar with his form of combat.
Holmes studies his opponent and preconceives combat movement then hits vital spots that cripple his opponents.
In regards to Holmes discovering Batman's true identity? If given enough time with the right resources it shouldn't be difficult for him to ascertain that Bruce's return to Gotham coincided with the appearance of the Batman. And working on that.
It's just a matter of who gets to who first. (Cue Terminator theme music)