My apologies Merc.
To be totally correct it was the Severnoaks Railway Company who built the line to the Bat and Ball. I have not got a clue as to why they stopped there but I am guessing at money problems. The LCDR leased the line before absorbing it in 1879. The connection to Tubs Hill opened in 1869 and trains ran from Tubs Hill to a platform at the Otford Junction but when Otford got a station these were run to there, this service ceased in 1880. From 1885 to 1898 there were no passenger trains run between the two stations.
I am sorry that I got it wrong but hope that the above clears the mess up, having spent a nice half hour with C.F Dendy-Marshall "History of the Southern Railway".
S4.