You can try, but I have my doubts. I mean, you shave that hair. It's under the skin. I'm not sure how well it would work, if at all. And even if it does work, you might have to do it again every day, because, well, the follicles still produce dark hair.
I think make-up is your best bet. A palette of concealers (or whatever they're called) of various colours like my sister has, and you take the contrary colour of the dominant one in the shadow on your lip - probably red or something. You put some on, and then you put some foundation over it, and the colour should be more or less neutralised if you've done it well.
I haven't tried it on a beard shadow, but when she put make-up on me once (and not much of it!) my sister could quite effectively neutralise the red of my irritation/acne with the green one, and her foundation isn't even one designed for heavy duty camouflaging.