What it puts me in mind of is the common Jewish expression I don't know from. (Which just means, I don't know or I don't know about.)
The Jewish Language Research Website has a section on Yiddish and English that says:
A number of Yiddish idiomatic constructions have also entered colloquial English, such as the pattern I don't know from ___ (ikh veys nit fun ___).
So, having heard I don't know from in the past, it doesn't sound unusual for me to hear I don't care from in the same way.
My guess is it was not a mistake at all but a deliberate play on the more normal Yiddish English expression. Also note, as per a comment to the other answer, Billy Wilder, the director of The Lost Weekend, was Jewish.