I think you are confusing the observer effect and the uncertainty principle. Schrödinger's cat is about the former, not the latter. Also, a point could be made about the fact that, according to general relativity, energy does bend space-time.... although one could then make a counter point about the fact that the center of gravity could have changed?
Anyway, I think Griffiths answers your question when he writes, in "Introduction to Quantum Mechanics":
The cat is neither alive nor dead, but rather a linear combination of the two, until a measurement occurs—until, say, you peek in the window to check. At that moment your observation forces the cat to "take a stand": dead or alive. And if you find him to be dead, then it's really you who killed him, by looking in the window. Schrodinger regarded this as patent nonsense, and I think most physicists would agree with him. There is something absurd about the very idea of a macroscopic object being in a linear combination of two palpably different states. An electron can be in a linear combination of spin up and spin down, but a cat simply cannot be in a linear combination of alive and dead. How are we to reconcile this with the orthodox interpretation of quantum mechanics? The most widely accepted answer is that the triggering of the Geiger counter constitutes the "measurement," in the sense of the statistical interpretation, not the intervention of a human observer. It is the essence of a measurement that some macroscopic system is affected (the Geiger counter, in this instance). The measurement occurs at the moment when the microscopic system (described by the laws of quantum mechanics) interacts with the macroscopic system (described by the laws of classical mechanics) in such a way as to leave a permanent record. The macroscopic system itself is not permitted to occupy a linear combination of distinct states. I would not pretend that this is an entirely satisfactory resolution, but at least it avoids the stultifying solipsism of Wigner and others, who persuaded themselves that it is the involvement of human consciousness that constitutes a measurement in quantum mechanics. Part of the problem is the word "measurement" itself, which certainly carries a suggestion of human participation. Heisenberg proposed the word "event," which might be preferable. But I'm afraid "measurement" is so ingrained by now that we're stuck with it. And, in the end, no manipulation of the terminology can completely exorcise this mysterious ghost.
The takeaway from the above paragraph is that the experiment is about the fact that there doesn't need to be a sentient observer for the "measurement effect" to occur.
I'd like to add to what Anna wrote:
it doesn't matter if you're certain or uncertain about whether the cat is dead: whether or not you weigh the box, the cat is not in a superposition of states, because the actual "measurement" was made by the Geiger counter, not by the human observer.