well if batman can't physically track him without the spidey sense going off he can probably still bug (no pun intended) peter's room without the spidey sense going off right? He doesn't go all tingly whenever he walks into a department store (security cameras). Other clues are enough to make bats at least have some spy equipment at his place.
(sorry i don't actually read much spiderman)
some times the spidey sense goes off when he is being spied on even by tech sometimes it doesnt it is all a massive inconsistency when it comes to him being followed and spied on.. it really comes to the intend to do harm to spidey i guess.... cause some heroes have followed spidey and spidey never knew it..
like to point out that the punisher was able to track and spy on spidey with the sole intent to kill him and his spider sense did not go off until the bullet was fired..
also wolverine has also ninja stalked spidey and logan isnt no steve rogers and by that i mean he aint so nice and spidey should be on edge around him anyways SS should have gone off 😇 😗
bats has his stupid Buddha monk training to help not only block his mind but his intend.. he controls his emotions and normally blocks his thoughts so as not to broadcast.. its very basic and rudimentary but its bn enough to allow him to become a void to passive mental scans like MM not that bats could hope to hide himself if MM actually focuses on him but for stealth purposes and quick scans its enough..
the spidey sense is suppose to be psionic in nature at least it was till ironman completely discredited it as such and said it was pheromone based.
plus he has psi blocks in his cowl as well... bats is always prepared
If Batman and Spidey lived in the same Universe I imagine Batman's distinction to Spidey would be being the only "normal" human to be able to sneak up on him. Because that's what Batman does.
That being said it depends whether Bruce meets Spidey as Peter Parker or not. If he meets Pete then he could figure it out on his first encounter but if not it might take him a while. But as said earlier he will eventually find out.