If no one knows it off-hand, I can make the effort later today of going through all the DJH/DH/DTNG eps and counting to figure it out

Actually do you know for sure he reached 200? If the list of characters' 100th appearance is accurate, he'd have been short by 3 episodes. (I'm pretty sure the list shows first the episode where the character reached their 100th appearance and then the number of episodes total the character was in). Unless it hasn't been completely updated, Archie would have only been in 197 episodes.