The following is based on Fallout 3 cut content and has not been confirmed by canon sources.
The Little Leaguer perk may have been intended to be a more specialized perk. Fallout 3's resources contain a Form List named "PerkLittleLeaguerWeapons"; this naming scheme is used by other perk-related Form Lists to define the weapons that the perk covers, for example "PerkPyromaniacWeapons" being used by the Pyromaniac perk. The list has only one entry: the baseball bat; it does not include the two cut unique variants of the baseball bat. The shortness of the list may be intentional, but may also be related to the late-development removal of the Throwing skill.
End of information based on Fallout 3 cut content.