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Very nice, Antoine!

The first clue for me: curvy constellation boundaries. Those were replaced by modern IAU coordinate-following boundaries c.1930. After that my analysis was quite similar to yours. Markab, as you note, is critical. It's just a little bit beyond 23:00 RA (=15° SHA for celestial navigation). I also used a few other stars including β Piscium and 25 & 26 Ceti. My result is 1915.0 +/- 2 years, which is basically identical to your results, and maybe your error bars are more honest! :)

There is a group on Facebook that I am sure many here would enjoy: the Vintage Astronomy Books community, created and managed by Richard Sanderson (Ian Ridpath regularly comments in that group, too). I posted this same imagery there, and a couple of people pointed out that Ball's "Atlas of Astronomy" is an excellent match. The first edition was printed in 1892, but that doesn't fit the precession evidence. Although it's possible, I can't imagine any star chart publisher in 1892 thinking ahead over twenty years for a precession epoch. A later comment suggested Ball's 4th edition published in 1925. I'm wondering if there was a 3rd edition right around 1915 or 1916??

Frank Reed

PS: The film is "set" in 1955 (-ish) but it's a surreal comedy so the lab coat on the astronomer Dr. Hickenlooper played by the exotic Tilda Swinton is just for style (Tilda Swinton was also fantastic as the White Witch of Narnia in 2005 and last year as a folklorist in love with a 3000-year-old Djinn).