So as things pass through the portal, conservation of momentum is either preserved or it isn't, with respect to a constant observer. What happens as they partly enter the portal in both of these situations?

If momentum is preserved, and they have zero momentum going in to the portal, then they are motionless as they exit the portal. There is nothing to cause your hand to move out of the way for your arm. Scaled down to the atomic level, you become a paste.

So you say that your hand moves out of the way because it is connected to your arm. The fact that it moves out of the way fast enough to make room for your arm means that it has velocity, and therefore momentum. The momentum means that it (and you) would get launched into the air, but conservation of momentum was violated.

There is no scenario where you exit the portal motionless but intact.