I have VERY recently gone through this.
How long have you had the puppy? Echo was a TERROR in this department for quite a while after I got her. She went through this earlier than your puppy (she's still a little younger than him), but your puppy's actions sound SO much like hers. And your reactions sound SO much like mine were.
Are you sure he has to go out to potty, or does he just want you to let him out of his crate so that he can be with you? Echo cried and screamed all night for quite a lot of nights when she was first getting used to her crate. She had never slept alone when we first got her. She wasn't used to being away from her litter and mother. It was a scary transition for her, I'm sure. At first I gave in to her crying and let her out. Then I tried leaving her in her crate despite the cries, but tried to calm her by talking to her.
I received some WONDERFUL advise on the forums to completely ignore her. Don't talk to her or acknowlege her in any way. It was SO hard to do, but after only two more nights of screaming, she miraculously stopped. Since then, she's done GREAT in her crate. We've had her now for a little over a month, and she's completely comfortable in there at night, and sleeps through the night. With the exception of just a couple of mornings (when she really had to go potty) she gets up only when we're ready to.
By the way, we DID set an alarm and take her out a few times a night to potty until her bladder strengthened. It could be that your puppy just doesn't have the bladder to sleep through the night. You could try setting the alarm for a couple or three times during the night. Take her out and immediately back to the crate where she gets not attention for whining afterward.
Anyway, I realize that your puppy is older and is probably used to the crate by now, but this all just sounded so similar to my recent situation.
I know it's VERY DIFFICULT and you're losing valuable sleep, but if you know that he is just wanting out and doesn't really have to potty, stand your ground. Ignore him. It is heartbreaking (and annoying), but the reward is amazing after he learns that he must remain there until you're ready to get up.