So ... first, yes, I'm using the V15 III (with Jico SAS stylus), so I'm excited to see how it sounds.
Turns out the guy from SME was wrong. The weight I have is what shipped with the arm originally and what the owners manual shows in use. The pics in the manual actually show the arm being set up with that weight and with a Shure V15 III, too.
I found a place in England called Charlton Analogue that sells SME arms and parts. The guy there noticed in pics I sent him that the black piece that protrudes out the back of the bearing assembly, which the weight screws into (it's called the spider) was a little funky. This piece is actually supposed to slide into the weight as you rotate it toward the arm. Mine was splayed outward a bit preventing the weight from being rotated all the way in -- by a good 3/4 inch. Thus, even though it felt to me like it was as tight to the arm as possible -- because the spider was stopping it from going any farther -- it actually had quite a way to go.
The spider has these little splits in it that make it a little springy. I was able to squeeze them together with needle nose pliers and with the help of a tiny dab of white lithium grease, I was able to get the spider into the ring around the screw on the weight, and voila! The weight now twists all the way flush to the bearing assembly -- and I can balance my arm.
All it cost me was about 8 hours of sleep last night trying to figure out what the hellI was doing wrong.
Thanks to everyone who chimed in!