My headcanon for this team is that they're officially called the [New Haven] Base Ball Club, and then before a game in the Union Association the opposing manager says "they'll crumble like an old doll", basically the Athletics' white elephant origin story. So they end up with a messed up old doll as their mascot. Around the turn of the century they rebrand as the Dolls using cutesy imagery of the time that would look unintentionally creepy to us now, like dolls in the outfield smiling too widely. In the 40s and 50s they probably have a creepy sexist rebrand as the Dolls. Basically this
Cardinals pinup illustration, although I prob won't bother making that.
Then they finally settle into a modern identity, proudly back to that weird slumped dover doll, kind of like the Phillies, Dodgers or Mets, with their strange archaic names that are so permanent that the just make sense. Like Juventus, "the Old Lady", or even something like the Toledo Mud Hens. There might also be an updated "active baseball doll" character, kind of like Mr red, that oscillate between creepy and cutesy.
The Old Dolls is just such a great idea. Please tell me more about what your league looks like that it can accommodate a name like that!