First off, THANK YOU!
You guys have been extremely helpful to this novice.
tl;dr Cool story, new chuck, all better.
I've haven't had a ton of time lately (life gets in the way of play), but I wanted to leave an update to where I'm at.
After checking and re-checking everything I was still stumped, but I think the 3 jaw was the culprit.
By coincidence I met a very nice retired machinist and was talking with him about a totally unrelated subject. By some miraculous stroke of luck we were right next to his storage unit and he invited me over to see his stuff. When he opened the overhead door it might as well have been the Pearly Gates. It was packed with old, some ancient, tools and machines. I must admit that I didn't even recognize %90 of what was in there or could even guess at it's purpose, but I was elated.
During our chat I happened to mentioned my problem and asked if he had any advice. He asked some followup questions about the machine, what I've tried, etc. and came to a similar conclusion. Get a 4 jaw chuck. Then he thought for a second and said "... let me see something". You can imagine my surprise when we walked across the row to a second unit, again, chock full of goodness. He started digging through a crate of discarded things and emerged with a rusty 4 jaw chuck complete with back plate and years of grime. It was exactly what I needed and, seeing my excitement, he offered me a generous deal.
Got it home, cleaned it up, went to put it on. It didn't fit.
The first thread would grab but something was wrong so I re-checked the size and threads, re-cleaned the spindle and back plate threads, still wouldn't go. I'm still unsure what brand the back plate is because the logo is worn off, but the chuck is a Pratt Burnerd so I called them up. Those guys are amazing! After a good talk with a very patient technician he said they do free assessments and gave me pricing if a new back plate is needed, but he'd call me first. I packed it up and shipped it off. About a week later I got a call from the PB tech. All done, just need a credit card to ship it back.
Wait, what?!
Turns out that the thread depth was just a hair too tight, so he took it out to the shop and chased the threads. No charge.
He shipped it back to me and when I opened the box I thought he had accidentally sent me the wrong thing!
This chuck looks brand new!
Turns out that during their assessment the Pratt Burnerd techs tear it all the way down, clean everything REALLY WELL (did they polish this?), then measure, lube, and put it all back together. They went above and beyond on this one and I could not be more satisfied.
The new 4 jaw chuck fits and looks wonderful. Made a few cuts and the wobble appears to be gone. I don't have any of the right size of round stock right now to duplicate when I was getting issues, but it already cuts and looks better.
I know this is a big wall of text but there were just too many people being helpful and awesome that I had to share.
Thanks again! I'm sure I'll be bugging you about some other rookie mistake soon.