Why is it all or nothing?
How about middle ground? Make a raise but not all-in - say to $1500. It's significant relative to the bet and his chip stack but still leaves you enough chips to play with.
BB might smooth-call but more likely he goes over-the-top all-in with those Aces. Then it's an easy fold and at that point you would be in great shape to cash and not be out of winning the thing.
Remember, they were only a middle pair.
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