Running “filmsy” through genealogybank produces over 600 hits, but the vast majority are OCR errors. In some cases I can’t tell. A few cases use what is undoubtedly “filmsy” to mean “flimsy.” Some of these are undoubtedly typographical errors, as “flimsy” is used in the same article. Others use the word only once, and so are inconclusive. I don’t find any article that uses “filmsy” for “flimsy” multiple times.
“Filmsy” is, however, undoubtedly a real word. It occurs multiple times in the context of fabrics, often in advertisements for dry goods merchants. Here are a few examples:
“For upholstering there are rich and heavy silk brocatelles at $10 a yard, while the filmsy fabric in imitation of India silk is as light in price as in weight.” Springfield, Massachusetts, Republican 12/21/1889
“The bridal veil, confined by a wreath of orange blossoms fell in filmsy folds over the graceful train.” Wheeling, West Virginia, Register 6/25/1890
“12,00 yards of the finest and most filmsy Lawns, 30 and 22 inches wide, in varying light effects and in all sorts of dark styles.” Washington, DC Evening Star 7/14/1898
“Dressing Sacques of every known material, from filmsy silks to the soft, warm eiderdown flannels.” San Jose, California, Evening News 10/27/1900
“The first purpose of the scarlet Socks Society, as explained by Mr. Koch to his assemblage of youngsters is to have the Brooklyn school children wear warm hose. At present he things they wear a too filmsy stocking. Many, he believes, are kept away from school because they have not warm hosiery to wear.” Bellingham, Washington, Herald 5/10/1904
Here is a metaphorical use where I think “filmsy” works better than would “flimsy,” so I don’t think it is a typo:
“The greatest of physical paradoxes is the sunbeam. It is the most potent and versatile force we have, and yet it behaves itself like the gentlest and most accommodating. Nothing can fall more softly and silently upon the earth than the rays of our great luminary–not even the featherly flakes of snow, which thread their way through the atmosphere as if they were too filmsy to the demands of gravity like grosser things.” Alexandria, Virginia, Gazette 12/2/1871
There is semantic overlap between calling a light-weight fabric “filmsy” and calling it “flimsy.” The late 20th century examples could be taken either way, so might be typos. The latest example I find with clearly positive connotations, and therefore not likely to be a typo for “flimsy,” it this:
“The Dior evening styles are elaborate but include many numbers in soft flowing chiffon with waving ostrich feather hemlines or sash ends or cuffs. Soft filmsy crepes and chiffons are treated to long sleeves, round necklines, and flowing circular skirts.” Baton Rouge Advocate 7/26/1968
So it seems to have been a real word, current in the late 19th and early 20th centuries, referring to film-like fabric.
This is, of course, unrelated to the purported etymology of “flimsy.”