OK, so Bruce had his very own Batcave installed on Arkham Island, because basically he "anticipated" the events that happened in the last game going down. But where's he gonna hide it this time? Wayne Manor is several miles outside of Gotham...
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There's catacombs hidden all under Gotham. Several have to hold gadgets. And at the very least, there's Wayne Tower.
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He has the batplane, you could just be picked up everytime you wanted to go to the batcave
^Yeah that. It's not like Batman is walking to Wayne Manor.
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I would doubt you would frequently return to the Batcave, so you might just fast travel there in the Batmobile once or twice.
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Batman has several "emergency" batcaves set up around Gotham for re-supplies and network access to the Batcomputer.
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Extreme is correct, Batman has several Satellite Bat Caves around Gotham, for example the one beneath Robinson Park. It's discussed in the "No Man's Land" arc and others. Not to mention there are several facilities hidden in Wayne tower that are used by Batman when travel back to the Bat Cave is inefficient or not possible. Also as depicted in Nightwing, Bruce has several safe houses set up within Gotham, no doubt they would be equipped with holdout supplies and facilities, and a secure network link to the main Bat Computer @ Wayne Manor.