The weaving of the head is due to sight. Even when you think the rat can't see you, they are still trying to make out what's around them. I have a few who do it quite often, and I've found that more skittish rats do it more often, or when they are scared. I believe it's that they're not confidant with their surroundings to not want to see what everything is, if anything is lurking and waiting to pounce on them, etc. 99% sure that is it, no brain tumor or anything. I went through the same thing when I had my first boy that weaved a lot... Thought for sure he was blind or something. But nope, just more prone to weaving.

Red and ruby eyed rats do it most of all, but again, I have found that more skittish and insecure rats do it more often. I've had red-eyed rats that don't do it at all and are very confidant rats, so that's somewhat what my theory is based on. Even once you get them to the point where they are much more relaxed, they will "regress" when extremely scared, whereas other rats in the same situation won't. One of my boys was a snake-reject and though he's 150% better than he used to be, in a situation where he freaks out, he will start weaving. Otherwise, he hardly ever does it anymore.
My suggestion for you is to take him out of the cage with a towel. This is what we do with our cage aggressive boys, and with some of our rescue rats. Compared to the glove-method, this seems to keep them much calmer. I have yet to have a rat attempt to bite when picked up with a towel around them. Of course, I don't guarantee anything.

It's something to try. If he's otherwise fine once he's out of the cage, then it's definitively cage aggression. There is probably something that is bothering him (some rats just plain don't like people messing with the cage), or he may have had an experience in the past. I'd suggest neutering, but you say he's already neutered. I've had success with switching cage aggressive rats to a new cage entirely, though that may just be my guy. He's not show any interest in biting since moving. Or, if you free-range in the same room as the cage, you could make it so he can come out on his own terms. That may solve the problem of getting him out.
I hate to see cage aggression as a reason for putting a rat to sleep. We have a few cage aggressive boys as personal pets, and as I said, in our rescue. They are otherwise great animals and sometimes, you can get past the cage aggression. Do you have a rat rescue nearby that may be able to handle cage aggression? If you were nearby (I don't know where you are), we certainly would take him in. I just hate to see him PTS for it.