WiS claimed in a Dev Chat once that the LH and RH specialists were working exactly as intended, which is ridiculous considering how infrequently they get brought into the game. There should at least be a separate setting for them so you can choose how aggressive you want to be with throwing them out there; I have a feeling that the current setting is that they only come in for high-leverage situations against one of the team's 2-3-4 hitters or some similar outregeous criteria (not based on evidence, since I gave up trying to figure out the specialists a long time ago).
My somewhat unhelpful advice would be to either (a) forget about the specialist and try to pawn him off on another unsuspecting owner who uses specialists, or (b) keep him around and enjoy the 12 innings per year you'll get out of him (which will, to be fair, be helpful innings). At the moment I really have no idea how to get more usage out of them, save for increasing their pitch counts, but that leaves them open to having to pitch to batters that don't fall into their specialty. I will say that you could change his "inning available" to "Any" which will allow him to get brough into more games.