OK. so the output is a digital signal between ground and 1 volt with a varying duty cycle. The problem being that it may not get all the way to ground.
Since you have +/- 15 volts to the op amp why not just increase the gain a bit so the output swings below ground?
IIRC, last time when I tried to do that, the driver reverses the polarity and increases the power output of the laser (the same as how it varies for positive going signal above zero). But, I will try it again with the new driver, but I expect the same.