However, the example has been so overgeneralized as to make me cringe. In particular, many editors who work with Chicago blithely ignore the general guidance that precedes this rule and gives it context. An example of such a permanent compound is long-term. Our approach is long term. This is not consistent with the reasoning presented in Chicago, and is fussy to the point of silliness besides.
Dictionaries list long-term as an adjective, and spell it with a hyphen.
So the hyphen should stay with it wherever it goes. Even if you steadfastly refuse to believe this word is a permanent compound, there is no excuse for the inconsistent hyphenation, because long term approach is unambiguous. Additionally, readers will note inconsistent hyphenation if it crops up to frequently. If anything, it is the hyphenation flip-flop that is likely to distract and interfere with easy reading. These are established in usage the same way permanent compounds are, and should be respected.