I wouldn't let him out when he starts complaining at 4 am. I'd set an alarm for half way through your sleep, wake him up, take him on a leash to potty so he doesn't get to fool around, and then put him back up with a tiny treat upon going in the crate. So if you go to bed and 10 and wake up at 6... pick up food after dinner, no water after 8.30, go to crate at 10 with a tiny treat to reward going in the crate. Wake up to the alarm at 2 (or early enough that he won't have woken up on his own), leash potty and tiny treat when he does his business, right back to the crate with another tiny treat. Go to bed and wake up at 6 am, take him out, and start your day. Don't make a big deal when you wake him up and let him out during the night... it's all business. Don't let him out when he cries. My guess is that he legitimately needed to go out the first couple of times, but then he learned that screaming got him out of the cage. Let him out on your own terms before he has a chance to cry, and then you know if he does it's not because he needs to go out. You will probably have to suffer through some crying for awhile.
Of course, make sure during the day that you are crate training him properly. That doesn't mean just putting him in there for a couple of hours each day to figure it out. If you haven't read the links on this forum about crate training I would highly recommend them. Make sure that the crate is a happy, joyous place with lovely gifts. Feed all meals there, give tons of treats, give particularly special treats and toys there. Get him comfortable being in the crate with the door open, and shut the door for very short periods of time with you standing there and reward when he doesn't cry. Gradually build up time to where he can be in there for a couple of hours.
Went through a similar thing with my puppy. He initially cried when we put him in the crate for bed, and then he would go to sleep. I'd wake up at some point to take him out (on my schedule, not when he woke me), and when I put him back up he would cry again. He quickly realized that the midnight potty was just going to happen and that crying wouldn't make us appear before morning. The cries lessened significantly each night, and soon he'd just walk back into the crate on his own at night. During the day was a bit more of an issue because we didn't have a consistent schedule, but we worked that out fairly quickly as well. At my dogs' ages and levels of crate/house training I know if they're crying in the crate that they feel bad and need to go outside NOW