"In its mother's fat" makes even less sense than "in its mother's milk", so I'm slightly surprised that isn't the tradition.
More seriously, I'm glad to hear that grammatically "on its mother's milk" makes sense. At worst the alternate interpretation I've put forth is a broader version of the literal meaning of the traditional interpretation. Of course, if one takes the flights of fancy concerning the separation of meat and dairy found in kashrut as your basis, my alternate doesn't support that. But then if I were a Jew I'd lean towards being a Karaite anyway, so that doesn't bother me.