It is indeed ugly, and technically, I'd say you're within your rights to write your sentences that way. Having said that, I'll side with George Bernard Shaw's criticism of apostrophes as "uncouth bacilli". I think your sentences are replete with uncouth bacilli, for the sake of someone's (whose?) propriety.
I would not jump through the hoops of punctuation as you've done. I would hope common sense would allow rewrites as follows:
He said, “I thought I heard Judy say, ‘“Gone with the Wind's” cast was unparalleled.’”
Joe said, “'Gone with the Wind’s' cast was unparalleled.”
“Gone with the Wind's” cast was unparalleled.
“‘Gone with the Wind’s' cast was unparalleled,” Joe said.
Joe said “I thought I heard Judy say, ‘It was "Gone with the Wind’s’.”
There is no space between double and single quotation marks.
James Harbeck writes
Many apostrophes are really only there for condescension. Apostrophes do have one consistent function: The grammar griper brigade likes to use them as the tips on their cats-o-nine-tails. Theyre excellent tools for condescension. Dont tell me theres no classism in terms like "greengrocers apostrophe" (for apostrophes in plurals). Such agitation over a little mark that conveys nothing new — other, evidently, than, "This person doesnt know how to use apostrophes." Its a fashion infraction on the level of wearing white after Labor Day or socks with sandals.
Should the Apostrophe Be Abolished?
“England and America are two countries separated by a common language.” -George Bernard Shaw