[ImNotJohn] No, no rabbits involved.
[Tuj] As indicated in my response to Chalky, above, there are no felines (including domesticated cats) involved.

Maybe the other connection is a dog as well.

[cfm] Not "That dog won't hunt".
[Raak] Not "dog eat dog"; the other animal connection is not the dog, Canis lupus familiaris. Nevertheless, some dogged audience members do sit up and take notice.

[Raak] No, not Biblically proverbial.

Well someone had to say it ...

[Phil] Not a wolf in sheep's clothing... (yet a few audience members give faint nods of appreciation)
[CdM} The word "dog" does indeed appear on the card.
[ImNotJohn] No, the words on the card do not involve the idea of hunting, yet there is a tangential connection.
[Raak] Not a dog's dinner.
[Chalky] No bollocks involved.

Time for a recap, perhaps - The answer is a well-known phrase that contains the word "dog'; does not refer to cannibalistic dogs; there is at least one other mammalian animal connection, non-human, which is not the domesticated dog Canis lupus familiaris or rabbits, or any Felidae; the phrase is non-Biblical; perhaps tangentially connected to hunting, and that the phrase "wolf in sheep's clothing" evoked a couple of faint nods of appreciation, as though tracking in the right direction.

[cfm]Not a dog and pony show.

[Raak] Not a dog in the manger. (Remember, the phrase has another mammalian non-human animal connection in addition to "dog").