Battery voltage drop

You can not just ground it and leave it like that, becuase it'll burn up your battery (and quite possibly other electricals in the car)
I meant just long enough to make sure it is capable of charging.
If it is capable then you should hear it drag down the engine a little and the alternator sound will change.

If it does charge, then after checking all the connections and grounds I would replace the regulator.

I think if your regulator is working then yes you need to ground that second lug and leave it that way. It will do one of two things, it will charge normally if the regulator is grounded and working properly. If the regulator is bad then it will probably not charge at all. From the picture it looks like that regulator is a newer version but definitely the older style that uses the grounded lug on the alternator.

TrailBeast is correct that if you were using the newer voltage regulator (which you are not) you would ground one lug of the alternator TEMPERARLY to see if the alternator is working.