I was wondering if the expression "swing a cat" might be an amusing turn of phrase derived from "swing a cot", which was used to express a similar meaning (cramped space) in a similar, nautical, context.
A google search shows the expression used, e.g., by Elizabeth Simcoe in her diary "Sailing for the Canada" (Sept. 26, 1791: "I became giddy (seasick) as soon as I entered the ship and went to my cabin, an apartment just large enough to swing a cot, which I immediately got into."
And the 1825 East India Guide has this: "Should a passenger's circumstances enable him to hire a cabin, his comfort will be very greatly increased, even with barely room enough to swing a cot, or to put up a standing bed."
I realize that these aren't as early as some of the listed examples of "swing a cat". And my apologies if this is already a familiar (and long rejected) notion.