I found a couple more explanations of this phrase online at messybeast.com, a treasure trove of sayings about cats.
(1) ‘cat’ is an old Scottish word meaning a rogue; therefore, ‘swing a cat’ refers to hanging a felon.
(2) ‘to swing a cat’ was a nautical term referring to the difficulty of maneuvering a sailing ship (a ‘cat’, a compact merchant vessel) via mooring lines in a confined space such as dock or narrow waterway.
Partridge’s Dictionary of Slang and Unconventional English makes no reference to either of these meanings of ‘cat’, incidentally.
The problem with the cat in question being a cat-o’-nine tails, the most commonly accepted explanation of the phrase, seems to be that the expression was in use in the 1500s while the whip itself wasn’t invented until the late 1600’s. According to the Random House Unabridged Dictionary, a cat-o’-nine-tails is ‘a whip, usually having nine knotted lines or cords fastened to a handle, used for flogging. [Origin: 1685–95; so called in allusion to a cat’s scratches]’
Moreover, according to Mr. Partridge:
To be ‘cat-faced’ is to be ugly, from mid 19th century.
A ‘cat on a testy dodge’ is a ladylike beggar pestering ladies at their homes for money, ca. 1870-1914.
‘Cat-witted’ is obstinate and spiteful, ca. 1660-1930.
To feel as if a cat has kittened in one’s mouth refers to the after-effects of being drunk, from ca. 1600.
Along those same lines, shoot the cat means to vomit, as does jerk/whip the cat.
Therefore, to be sick as a cat is to be very sick indeed, most especially after overindulging in the grape.
Ah, the bad old days.