You can represent Batman as almost anything, actually - he's been an example of how broadly applicable splat thematics are on several occasions, at least once by
@Revlid. Solar, Lunar, Sidereal, Abyssal, etc.
And you don't actually earn a Solar Exaltation by doing something impressive. You get a Solar Exaltation
in your time of greatest need, and
then do something impressive
with it. Dace, Panther, the Bull... none of them were doing anything particularly exceptional when they Exalted. The Exaltation, drawn to their need, was what
allowed them to excel - Dace shrugged off his age and weariness, cut through the mercenaries and won the battle he should have lost. Panther stepped away from his life of vice and decadence with newfound purpose. The Bull rejected the law of his people that said he was too old to contribute anymore, survived the storm and went on to lead them. Arianna became a Twilight as a driven, brilliant woman in an institution that overlooked and ignored her, accomplished something they'd never been able to do by translating an ancient tome, and then gave them the finger and went off to seek adventure. Etcetera.