I don't think the food supply was an issue, the issue was how much of their food they could carry with them. The ship was probably well-provisioned, they just had to abandon it and take what they could carry.

My understanding was that Shackelton felt that taking the cat along (and making sure it had enough to eat, etc.) would have endangered the survival of his men. Of course, my knowledge of this is based on the Kenneth Branagh mini-series, which is not exactly an authoritative reference.