John, you're right about a lot of the Tin Pan Alley songs being not PC.   ("Negroes" might have been considered non-PC by some then; the term "darkies" was considered a euphemism by some, and was even used in songs like "That's Why God Made Darkies.")   But I digress.

As for this Berlin song being off-limits, I think it depends on your audience and who you are.
The song is not about a Jewish merchant contrasted with "goyim." It's about a businessman's dealings with others within his own ethnic community.   Janet Klein, being Jewish herself, is probably not going to be perceived as PI when she does this song, complete with strong Yiddish accent.   

My concern is that, while I've often done accents and played characters in theatre and as a musical performer (and am routinely complimented on the credibility of my accents by people whose native languages I sing in), since I am a "shiksa," some people in the Jewish retirement communities where I perform regularly might take offense.    Before I add it to my repertoire there, I'll probably run it by the activity directors and maybe a few residents.
(I actually did share it with a retired couple yesterday after I had finished a program at an upscale Jewish retirement residenct.   They were not familiar with the song and the woman, Margie, was very surprised that she had "missed an Irving Berlin song."   They both seemed to think it was funny, but I thought I detected a scintilla of reservation on their part.   Further research is needed.