There are two things going on here: 1) the labels used in Lightroom are different than those used in Bridge. (Don't ask me, Adobe didn't ask me either :>)) So, it ends up being one is talking one language while the other is talking another language.
2) The white happens because you (apparently) changed the name of the labels. So, if you changed the "Red" found in LRC to match Bridge's "Select," than that's not what is expected and it defaults to white. In fact, if you changed the name of all of the labels, they'd all end up being white.
Seriously, LRC and Bridge are mutually exclusive applications. I use both daily but the files that I use in LRC are not the files I work on with Bridge. One is best served by both if you do not overlap your work. You are witnessing one of the many reasons why. (Yeah, I know that this is not what you want to hear, but it's for the best.)