The cross references to other titles by the same author in Stephen Barr's books quickly revealed that he is also the author of the following:
Stephen Barr, A Miscellany of Puzzles: Mathematical and Otherwise, New York: Thomas Y. Crowell 1965. Republished as Intriguing Puzzles in Math and Logic, Dover 1994.
Stephen Barr, Second Miscellany of Puzzles: Mathematical and Otherwise, New York: Macmillan 1969. Republished as Mathematical Brain Benders, Dover 1982.
Stephen Barr, Puzzlequiz: Wit Twisters, Brain Teasers, Riddles, Puzzles, and Tough Questions, New York: Thomas Y. Crowell 1978. Republished under the same title by Harper & Row 1990.
The most important clue provided in the question was Barr's friendship with Martin Gardner. Gardner makes an appearance in blurbs for Barr's books, while Barr is mentioned multiple times in Gardner's publications, with indications that besides having an interest in topology as an amateur mathematician he was also an avid creator of puzzles. Two examples:
Martin Gardner, New Mathematical Diversions: Revised Edition, Mathematical Association of America 1995, p. 37:
- Barr's Belt
Stephen Barr of Woodstock, New York, says that his dressing gown has a long cloth belt, the ends of which are cut at 45-degree angles [...]
Martin Gardner, Martin Gardner’s 6th Book of Mathematical Diversions from Scientific American., American Mathematical Society 2020, pg. 13:
Now, it is true that at the time he wrote this apparently no one had tried to make a paper Klein bottle, but that was before Stephen Barr, a science-fiction writer and amateur mathematician of Woodstock, New York, turned his attention to the problem.
Given that Barr is identified as a science-fiction writer living in Woodstock, NY (a small town of some five-thousand inhabitants), one quickly finds various SF short stories by him, mostly published in Galaxy Science Fiction and The Magazine of Fantasy and Science Fiction, which I gather were premier American outlets for this genre in the relevant time frame.
Barr's science fiction writings range from the 1950s to the 1970s, for example: "I Am a Nucleus" (1957), "Callahan and the Wheelies" (1960), "A Sense of the Future" (1972). A comprehensive list can be found at the Internet Speculative Fiction Database. The Atlantic published two articles by Barr, in 1955 and 1961, and identified him as a
[...] free-lance writer who lives in Woodstock, New York.
Find a Grave is a good resource for finding information about dead people, and given the dates of Barr's publications, I considered it very likely that he was no longer alive, and looked there for someone of that name buried in Woodstock, NY. Sure enough, they have a picture of a relevant grave marker in Woodstock Cemetery, which shows that Stephen Barr was born in 1904 and died in 1989.
With all the preceding information in hand, I was finally able to track down a brief biography of Barr which confirmed the biographical data already found and adds new details:
Otto Penzler (ed.), The Black Lizard Big Book of Locked-Room Mysteries, Knopf Doubleday 2014, pg. 584:
Barr was born to American parents in England and grew up there before moving to the United States, possibly around 1930. He worked as a commercial artist and architectural draftsman before turning to writing full time in 1955.
From the fanzine Yandro, Vol. 15, No. 8, September 1967:
Remember that fuss about Stephen Barr allegedly plagiarizing stories
for fanzine submission? Well, he had an entry in CONTEMPORARY AUTHORS and
besides such things as his birth in England (in 1904) and present home in
Woodstock, N.Y., we learn that he was long a commercial artist and
architectural draftsman, turning free lance writer in 1955. He also
contributed stories and articles to Vogue, Mademoiselle, Atlantic,
Harpers, Playboy and similar such crudzines.
Google provides only a snippet view of Contemporary Authors, Vol. 1, Gale Research Company 1975, p. 39:
BARR, Stephen 1904 PERSONAL: Born 1904, in England, son of U.S. nationals; married Anna Keiley. Education: Attended schools in England. Residence: Woodstock, N.Y. CAREER: Commercial [...]
A 1940 civil list for the City of New York shows Anna Keiley Barr working as a social investigator at District Office 25 (212 E. 125th St. Manhattan) of the Department of Welfare, with a start date of Jan. 1, 1938.
UK records show that the birth of a Stephen Barr was registered in Oct.-Dec. 1904 in Uxbridge in Middlesex. This is a suburb of London. What is publicly available is a summary page; I have not found a way to access the actual birth record.