Dr. Fuller lived from 1608 to August 16, 1661. He wrote The Worthies of England and books with titles like The History of the University of Cambridge: from the Conquest to the year 1634. While these titles may not sound witty at all, his book of meditations Good Thoughts in Bad Times, illustrates this meaning of wit, which we also see in the quote about cats. The bad times to which Fuller refers was the 17th century English revolution. He was on the losing side, til the King was invited back, and he said of those years: "All that time I could not live to study, who did only study to live." Wit exemplified.
In fact, "Antithetic and axiomatic sentences abound in his pages." "Wit," wrote Coleridge after reading the Church History, "was the stuff and substance of Fuller's intellect". Thus the Encyclopedia Britannica about our Thomas Fuller, D.D.
I am at some pains to point to the possibility the above Thomas Fuller could have written our sentence, since the sentence appears in a book by a different Dr. Thomas Fuller, a medical doctor, who lived from 1654 to 1734. He "collected" proverbs, and published:
Gnomologia: Adagies and Proverbs; Wise Sentences and Witty Sayings, Ancient and Modern, Foreign and British (1732).
This book contains thousands of proverbs, and this Fuller claimed his work was the largest collection of such extant. While there is no reason to think he wrote the above proverb, he collected it. It does indicate the proverb was in circulation by 1732.
Of course it is quite possible that other people have attributed our first Fuller with the authorship because they have confused the two writers, lost in the mirror maze that is the web citation.