Yes, lying can be moral, so can stealing and killing, depending upon context.
It's totally different to genuinely feel that it's kind and right to tell someone they are beautiful, and to tell someone they're beautiful and not feel it on any level, and just actually feel hateful inside.Regarding your definition of what "fakeness" entails, I could spring a slew of quotes of definitions to prove lie, false, fake are synonymous. Your personal definition of what fakeness seems to regard whether the fakeness is overt or not. But, the action the person is committing (lying to your face when they internally hold another opinion), is it any different whether or not they roll their eyes to someone? I honestly can't see how it is.
Yes, not genuinely "feeling it" is fake.