Admittedly, as presented, it is slightly flimsy in an 'evidence' sense - but it makes for a better scene. Blake is shown to be a skilled and persistent detective in the film. If he first had an inkling when he was a child, it's more than likely he'll have done some follow-up and perhaps noticed Wayne went into recluse mode at the same time the Batman disappeared.
There are also plenty of instances to make the connection if tenacious and skilled enough, as Blake is clearly presented to be:
- How did Batman show up so quickly at the Dent fundraiser the Joker crashed, for instance? That would have been reported in the news.
- At the press conference where the Batman was to reveal his identity and Dent did instead, it's more than conceivable his presence would have been reported (''To a crowd of police officers and civilians, including ... blah'') - why would he be there in person? He's simply a fun-loving playboy in the public eye at this point.
- Police reports of the direction the Tumbler was fleeing in Batman Beginsbefore they lost it? The direction of Wayne Manor.
- The first emergence of Batman was at the same time as Wayne's return from a seven year absence. Both were heavily reported.
- The fact he has so much equipment means the real identity of Batman must be a man of means or with powerful friends. Wayne is both.
Taking a further look, combined with his (later established) good instincts leads him to be fairly certain he's right. And there are enough clues for him to put it together if skilled and observant enough.
However, explaining all that makes for a crap piece of character interaction and doesn't give any depth to the way Wayne is perceived by others. It's better to go with what first got him thinking it might be possible. The way it is written is better, perhaps at the cost of spelling it out fully.
N.B. He's not sent by Gordon. He goes through his own initiative in the wake of Gordon's injuries to encourage the return of the Batman.
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