How about - order should be preserved if we shift the zero-point of our happiness measurement. That seems pretty common-sense. And yet it rules out total utilitarianism. (2,2,2) > (5), but (1,1,1) < (4).

The usual definition of "zero-point" is "it doesn't matter whether that person exists or not". By that definition, there is no (universal) zero-point in average utilitarianism. (2,2,0) != (2,2) etc.

By the way, it's true you can't shift by a constant in total utilitarianism, but you can scale by a constant/