Ive raised this question previously, as my character has a patch around his eye, like a birthmark, which wouldn't make sense being flipped from the left to the right side. Ive asked this during one of the live streams and was told people wouldn't really notice.. so not much help, so I had to come up with my own solution.
What I did was duplicate the character. Named it "Character Mirrored", then I took that character into photoshop, and pasted the left eye to the right, and the right to the left side, saved it. Ofcourse now the character looks wrong in cartoon animator, but when I click the flip option it becomes right and that patch was on the correct side.
So that meant, I had x 3 of the same characters. x1 front facing, x1 side facing, and x1 mirrored version (for flipping) and that seemed to help my problem. I think that maybe with hair that might not be too much of a problem.
Conclusion, make a duplicate character and flip the features in photoshop.
Hope that helps. But yes, it would be amazing if they had a feature where you could disable flip to certain sprites. it would be such a help. Ive asked this from Reallusion a year or two ago in the Wishlist, but I didn't get many votes in support.
Last Year by