I think the reference to Bucky's pet rat appears just after Halloween.
Having gone through the script which someone posted on one of the other threads here, I noticed there were lines about Gary having set up pranks to play- one of which included throwing a dead rat at a girl on Halloween night.
I'm assuming that Gary supplied the rat- or that you were meant to acquire it from somewhere, and was assuming that the day afterwards, Bucky noticed his pet missing.
Anyone else notice Edna talking about what she'd do if Jimmy didn't get the meat from Yum-Yum Market in time, and then the later bulletin board sign about Ms. Phillips' cat being lost? I assume that was a similar reference.