Of course you can have a "sympathetic" bigot. Easier said than done, but it simply requires writing a multidimensional character. Just take a look at, oh, Guess Who's Coming To Dinner, to find several sympathetic characters who first exhibit bigotry and who then grow to rethink their positions over the course of the evening.
Whether a character who has bigoted tendencies can also be sympathetic is a wholly separate question from whether such a character should be a regular. Not all Star Trek regulars are either villains or crewmen. Quark comes to mind as a regular who was neither. Plus, assuming that we are talking about a show with genuine character arcs, we don't need to presume that some character traits won't change over time.
Confining the discussion to crew personnel, granted it would be harder to pull off. On TOS, although he was not a regular, Mister Stiles exhibited bigotry against Mister Spock in "Balance of Terror," which simply goes to show that at that point (in the development of the show and/or in the history of Starfleet, whichever way you want to look at it) bigoted tendencies did not preclude someone from graduating from Starfleet Academy and getting top-drawer postings as a commissioned officer in Starfleet.
To pull it off, it would require, at least, a top-notch group of writers, showrunners willing to handle controversy, and the belief by the showrunners that it wouldn't negatively harm viewership. Not saying that's what they should necessarily do, although the greatest shows of television often didn't play it safe.
