OP, you are on the right track for questions... and a possible can of worms in seeking answers: the only 340 ATI damper listed in the link from 70aarcuda is for the internal/neutral balance setup.
IF the engine was balanced for the stock cast crank 340 balance setup, then there should have been a correctly imbalanced damper for a cast crank 340 on in the first place (not an ATI from the list posted), along with an inmbalanced flex plate (or something simulating the imbalance of the TC for the cast crank 340).If this was the case, you could pull off of the transmission inspection cover, slowly turn the engine, and look for either a 'bite' out of one edge of the flexplate, or a rectangular steel weight welded on the front surface of the TC near the outer edge. I would do this regardless of anything else at this point.
BUT, if you have much lighter pistons and rods than stock, then the 340 cast crank could have been balanced for neutral balance damper and flexplate/TC. (No 'bite' out of the flexplate nor any extra weight welded on the TC.)
SO, do you know what pistons and rods are being used? And do you know for a fact that the crank is a cast one? Even if it has a later 340 block, someone could have put in an earlier forged steel crank, and used neutral balanced damper and flexplate/TC.
Do you know the shop that balanced it? That would be the fastest way to know for what parts the crank was balanced.
Are you having vibration issues?