An issue came up while my group was playing Chain of Command where both the "Imperial Officer" version of Weiss and the empty AT-ST had damage on them. When Weiss entered the AT-ST the imperial player removed the damage from the "Officer Weiss" and only counted the damage that was on the AT-ST. Was that the correct way to play it, or should the damage have been combined, which could effectively kill Weiss outright if there is enough combined damage?

I think the Imperial player was correct in only using the AT-ST's damage. You would have to blow up the AT-ST to actually kill the man in it, even if he was already hurt. Just my thoughts.

Yes, the rebel players all agreed that is how it should be played. But are there any rules that address the question?

The elite officer card that is Weiss is no longer in the game, the damage on it is therefore not relevant.