Let's not get complicated here, the answer is very simple. Discussions of rod and/or piston weight muddy the water.

A '68 Charger 383 was internally balanced, so the converter is neutral balanced.

A 440 at this stage in life is as likely to have a cast crank as a forged one. In my experience, if one is looking for a decent used 440 the engine found will most likely have a cast crank. This is because the later 440s were not very high performance engines and the early ones have been hot-rodded preferentially over the years, making them scarcer.

Look at the Engine ID pad under the distributor. It'll tell you everything about how the engine was built. Dollars to doughnuts, there will be a letter "E" stamped somewhere on the pad. That "E" stands for cast crank.

I am now assuming the engine is a cast crank 440 that was swapped into the Charger in a simple unbolt-bolt manner. In this case the 383 torque converter was used with the standard MOPAR flexplate. Here is your problem.

The cast crank 440 requires a certain amount of external balance on either end of the engine. Assuming again that the harmonic damper was not disturbed, the rear end of the crank is now unbalanced and vibrates. A weight must be added to the rear of the crank.

Chrysler did this by welding weights on the converter.
Your friend can do it by swapping his stock flexplate with a special unbalanced flexplate. Converter stays as before.

B&M used to be the only ones who made this unbalanced flexplate, I think others do now, too.

Assuming again that the car was not a drag racer or daily driver, and it didn't see a lot of miles, the rear main bearing is probably OK, at least good enough to let the owner simply swap in the imbalanced flexplate and keep driving the car.

R.

Get the Engine ID number and post it. There are a few other possibilities that are not so likely, but the ID number will allow the Sherlocks here to ferret them out.