This is definitely a cool track but I don't see how it could've ended up on New Jersey, although New Jersey is still a glam metal album it still sounds more "mature" than SWW, Bon Jovi, and 7800 Fahrenheit.
You're Not My Lover could've replaced some tracks on SWW like Without Love, Wild In The Streets if they had written it in 1986. I'm not saying that Without Love and Wild In The Streets are weaker tracks than You're Not My Lover, i'm just saying that You're Not My Love could've been an alternative.
Edit: I just found this version
IMO it's better than what Dalton did.