At what point during the process would I drill the hole in the bottom of
the pieces for the weights? Initially, then turn 'around' the hole I
drill in it? I don't have a chuck to hold it with (Jet mini) so
logic(?)
says to do it before I turn. If I wait until its all turned and sanded
I'll scratch it up trying to hold it for the drill press or hand drill.

You could use Art's approach with the face plate and 3/8" dowel but then
convert it into a pin chuck. For a description of a pin chuck check Fred
Holder's page at http://www.fholder.com/Woodturning/tips.htm The description
of a pin chuck is near the bottom of the page.