The answer is not to dye it, but to use a french polish made up with the darkest shellac flakes you can find. That way, you can do the deed in the workshop and not the bathroom. You may have to substitute a long-bristled brush for the usual polishing rubber or mop, though.
Alternatively, find a builder's bucket large enough to accommodate your head, and use it dye both the beard and whatever hair you might still have at the same time. Remember to hold your breath whilst doing so, though.