Here's the deal:
LA flexplates are LA flexplates.
From the factory, the engines that were externally balanced had weights attached to the torque convertor to balance the motor. Internally balanced engines did not. No change to the flexplate.
The confusion comes in when you start talking aftermarket flexplates- B&M (and others) make specially weighted flexplates that allow you to run neutrally balanced (also known as internally balanced) torque convertors on externally balanced engines. These are not factory, though.
So, to answer your initial question, if you already have a 360-specific torque convertor (unmodified, with the factory weights still intact) and a factory LA flexplate, you're good to go.
As far as the harmonic balancer goes, if yours is in good shape, and the rubber isolation ring isn't rotted or the outer ring hasn't slipped for some reason, you're good to go.