I just did a quick reread of this thread and the mention of the phrase being in the 1936 movie "Come and Get It" based on the Edna Ferber novel. Edna Ferber was born in Michigan, lived for awhile in Chicago before moving to Appleton, Wisconsin when she was 12. She worked as a journalist in both Appleton and Milwaukee. Of COURSE she would have heard the phrase and used it in her writing.

I just googled the phrase and found this (longish quote following) at
World Wide Words

Q From Molly Walzer: A phrase remembered from my childhood: when one chokes, one might say, 'It must have gone down my Sunday throat!' That is, something has been inhaled rather than swallowed. I didn't realise this wasn't universally known until my husband questioned it. My parents are mid-western US in origin. Is something like this in general parlance?

A It's hardly common, to judge from the few references that have turned up, though it does still seem to be known today, and it's certainly American in immediate origin.

The two places in which I've definitively managed to track it down are both books from the early part of the twentieth century. One is The Lure Of The Dim Trails by B M Bower, dated 1907: "Hank was taken with a fit of strangling that turned his face a dark purple. Afterward he explained brokenly that something had got down his Sunday throat — and Thurston, who had never heard of a man's Sunday throat, eyed him with suspicion". The other is from The Eskimo Twins, by Lucy Perkins (1914): "The water went down his 'Sunday-throat' and choked him!".

Apart from this, I was at a complete loss. So I turned to members of the American Dialect Society. Douglas G Wilson suggested that Sunday here might have started out with its figurative meaning of "special" (as in Sunday clothes, for one's best) but that could have shifted to mean "alternative; other". It did so in the American slang expression Sunday face, which once meant a sanctimonious expression, but which took on a slang sense of the buttocks, that is, the "other" face. (Well, it does have two cheeks.) So Sunday throat just means "the other throat", which is clear enough, though anatomically inaccurate.

[Since this piece first appeared in the newsletter, subscribers have told me that they remember it from their youth, and that it also occurred as Sunday pipe and Sunday lane. It also transpires that similar phrases are known in Dutch and French, so the term may not be American, but one imported from Europe by immigrants.]