Composite video signals are usually AC coupled, so you shouldn't need to pull the output below ground. The receiving end will restore the DC level (using the sync pulse as a reference) if it needs to.
At the input side the signal may go below ground - or not, depending on the source. To cover all possibilities you should terminate the input with 75Ω and AC couple the signal to your op amp. You can then apply whatever bias is required to keep the op amp's output voltage above ground.
If you cannot ensure that the AC coupled signal's peak voltages stay within the op amp's input and output ranges then you may need to restore the DC level. This can be done with a diode, or with a sync detector and transmission gate that clamps the video signal to the desired reference voltage during the sync period.
Here's a simple DC restore circuit using a diode. The second diode and resistor provide 0.6V of positive bias to compensate for the negative voltage across the clamp diode when it is conducting:-
simulate this circuit – Schematic created using CircuitLab
Would the LT1006 be able to do this?
The LT1006 has low bandwidth and limited output drive capability. You should choose an op amp which is designed for video applications, eg. LT6205.