Like everyone says, there really is only one way to deal with the big stack bully, and that is to be patient and call when strong.
However, there are some other signs to look for. What is his pre-flop opening frequency (and for that matter is he 3-betting anyone?), what are his sizings and does he prefer late position or not?
a) If his frequency is high (60%, 80% more ...) then he is being a bit reckless. He is obviously wide and we don't necessarily need a prime hand to catch him.
b) If his sizings are large (like 3x 4x, etc) then he is being reckless. When we are near the bubble, 2x will get a good number of folds, especially from short stacks.
c) If he opens equally from any seat, he is being very careless.
On the other hand if his frequency is more like 40-50%, and leans more heavily to late position, and folds more UtG and MP, and his sizings stay reasonable 2x to 2.5x bb then he is being a pretty smart bully.
We would rather tangle with Careless Bully than Smart Bully.
Careless Bully is betting too often, to wide, too large and he is going to get caught, if not by us, by someone else - and if someone else gets him, that's great for us. If he loses half of his stack, he likely could stop bullying.
If we do want to call him, obviously we will need a strong hand. With Careless Bully though we can widen. ATo for instance will be well ahead of most of his range. Of course making this more difficult is the fact that if we lose, we bubble out. We've come this far and we just want our ticket, so it is a really sweaty situation. This is why if done right, bullying is a very good strategy.
Relative position is big here. If he is on our left, we will be opening into him unless we are on the BB (and we should not be limping with a crazy bully at the table unless maybe we trap him with AA). If he is on our right, we can let him open and decide if we want to call.
Also, any time that we are in a situation (bully or not) where we have a significant chance of stacking off, isolation will be a big deal. We don't want to be in a multi pot if we can help it. We know that Bully on the HJ is wide but if we call from the BT and the BB joins in, BB will NOT be wide. This will be bad. Even if we have AA KK we'd probably prefer to not see that. Although isolation might not be too difficult if Bully is making everybody fold 80-90%.
And finally, this is another huge factor. What are the stack sizes? Just how big is our stack and just how much bigger is his stack? If he has us outstacked 6 to 1 or 8 to 1 or more, assume that he WILL be putting us all-in at some point in the hand. If he's smart, sooner rather than later. Even at 4 to 1 he might, depending on his hand strength and how crazy he is.