The problem with this experiment is that a cat is a very complex system and the two particular types of states we are interested in (i.e. dead or alive) are very far apart in configuration space. It may help to imagine that we could rearrange configuration space a little to put all the points labelled "alive" on the left and all the dead points on the right of some line. If we want to make the gross simplification that we can treat the cat as a very simple system then this means that "alive" points are very close to the "dead" points in configuration space. In particular it means that there are significant amplitude flows between the two sets of points, that is significant flows across the line in both directions. Of course such flows happen all the time, but the key point is here the direction of the complex flow vectors would be aligned so as to cause a significant change in the magnitude of the final values in configuration space instead of tending to cancel out.

This paragraph doesn't sound as a correct approach. The main objection is that you don't need to classify all classical states as alive or dead to think about the cat experiment. It is sufficient to take one arbitrary alive state as the initial condition. Ideally, after coupling the cat to the poisoning mechanism, the system evolves to a superposition of two classical states whose labels are unambiguously "alive" and "dead". Speaking about flow crossing the line in both directions doesn't help understanding and even doesn't add realism: the distribution of states in complex systems is never uniform and the flow usually goes overwhelmingly in one direction across such macroscopic boundaries, as we know from thermodynamics. (By the way, it would help if you specified what do you mean by "amplitude flows between the two sets of points" more technically.)

In the original setting when the poison is triggered by a decaying nucleus, the cat doesn't oscillate between dead and alive in any meaningful sense. It is in an eigenstate of the alive operator initially and over time the projection to the alive subspace monotonously exponentially decreases in magnitude.

(If I can make a general suggestion for any arguments about quantum physics: Do it in mathematical notation first and then interpret the symbols. Quantum theory is perhaps the hardest discipline if one wishes to perform correct reasoning without math.)

Edit: An additional, although possibly not much relevant, problem is that you assume that alive and dead are properties that can be given to points in the configuration space. But that is not true: dead or alive are certainly sensitive to momenta of the cat's constituent particles, not only positions that are specified by the location in the configuration space. In other words, the "alive operator" doesn't commute with the configuration space labeling observables, unless you have selected a very special set of observables to begin with.