Jeri, "cat-stoppers" probably has something to do with the process of "catting" the anchor on them older sailing ships; there's the "cat-purchase", a set of pulleys and line used to draw up the anchor to the "cathead" and if there's such a line it probably needs to be secured to something, a bitt on top of the cathead and a "bitt stopper" is a length of rope used to more securely bind a line, say a cat-purchase line, to a bitt.

Now I don't recommend trying to explain all that the next time you sing that verse. However, some explanation is probably needed just to reassure the cat lovers in the audience that you are not making "free with the cat."

Cheerily,
Charley Noble