That's an interesting idea. Physically, we wouldn't have any way of testing whether or not there was a superposition of states. I think Schrodinger's idea was simply to tell us how bizarre the implications of the probability interpretation of the wave equation are (e.g. the idea that a cat, or any organism, could be both dead and alive at the same time). It is not actually an experiment that he thought out, since he was focused simply on the implications themselves. The question is: what would be the "screen" that would allow us to record the interference pattern of a cat being both alive and dead?