I am wondering which Overdrive Mode is best for grinding using Overside -> SP on a weapon. In addition the method best to employ when using said Overdrive Mode.
If you're going for SP grinding, there's a super easy way to go about it. The simple answer is to use Stoic for one, and Comrade for the other two; all the damage the character takes, even beyond their maximum, goes straight into Overdrive, and you'll get less for the others, but still a decent amount.
The slightly longer answer is to get everyone a weapon with Overdrive -> SP, and Triple Overdrive and Triple AP. Getting both will make it faster, but isn't an absolute requirement. One of your Comrade members should also have Auto Phoenix (if you're still at a fairly weak state. Optional if you're just trying to wrap up the sphere grid).
You then want to fight Master Tonberry in the arena. Your member with Stoic will attack, and the grudge will kill them. If you have Auto Phoenix on one of the others, you'll resurrect automatically. Otherwise just use a Phoenix Down. Either Comrade can cast Haste on the one that died, but should not attack. Repeat. If you can't kill the Master Tonberry before he gets too close, run. You'll still get a ton of AP, and will probably hit the max without too much issue if you do manage to kill him.
Personally, I like to set all three characters to use Ally; the increment is only 4%, but it can be endlessly repeated within a single battle, doesn't require you to do anything but take turns, and can be boosted by Haste. Kill all but one enemy in a battle, heal as necessary.
Another suggested setup is to have one character use Stoic, the other two use Comrade, and have the Stoic character use Provoke on enemies. Each time the Stoic character is hit, all three gain AP.
I don't have the rep to add a comment to supplement Frank's answer (which I've upvoted because I think it covers what you've asked well), the name of the best Tonberry here is "Don Tonberry", which does 10x the damage as regular / Master Tonberries. To unlock him, you need to capture one of every monster from the Cavern of the Stolen Fayth.
Also, you'll find that you go through spheres a lot quicker once you get into the swing of this method - my advice is to also unlock Kottos (capture one of every fiend from the Mi'ihen Highroad) and beat him after using one of the "Distil" abilities to get the type of sphere you want. If you overkill him, you can get 40 spheres each time like this.