I do think the kind of mission done in Doomsday Book would be relatively safe for a black man to do -- you'd have to play the "shipwrecked traveler" role, and deal with some general xenophobia, but not entrenched racism. But for the sort of "slip in and quickly infiltrate unnoticed" jobs that Lady Schrapnell wants you'd have to choose time and place carefully, and most of English history would be a no-go.