How did he know about the mountain lion?

This is brought up in the novel "Star Trek: Deep Space Nine: Fearful Symmetry" by Olivia Woods. (Clearly that plot hole had niggled at the author too!). Dax first speculates that this was a false memory that had been implanted at the same time they were applying the Cardassian plastic surgery to Kira. Ultimately, though, they deduce that it was a true memory, and Kira really had been a prisoner at Elemspur. Iliana had been intended to take on Kira's appearance and memories, and thus Entek knew about the memory as he was the one who copied the memory from Kira to Iliana. In the end Kira's memory of her time at Elemspur was wiped and she was set free instead, though.

Out of universe, it seems likely to me that this is a leftover of the original plot of the episode which had a more ambiguous ending about whether Kira really was a sleeper agent. According to the reference book Star Trek: Deep Space Nine Companion, the episode was originally going to feature O'Brien finding out he was a Cardassian sleeper agent, but the writer then had trouble in explaining how O'Brien's daughter Molly could be human, so the focus character was switched to Kira. In an interview with the writer on the DVD bonus features:

he says that "In my original version of the story, Bashir could not conclusively tell her whether she was Bajoran or Cardassian at the end." Leaving dangling plot threads like the fact Entek knew about the cat could further increase the ambiguity of the ending.