OK, today I had some extra time to work with the wheel and had some better results. I measured the circumference of each side of the wheel and they were about 3/16ths off. Took the larger side off, cleaned things up and put them back together and it spins much truer. Also really took my time and used a rubber mallet wherever I felt a bump between two assembled pieces, did the same on the other side and repeated until I had no bumps.
They need to include a rubber mallet with the wheel. You just can't leave it to chance that people are pushing things together well enough for the wheel to be true. I used a rubber mallet through my initial assembly, and again on my re-assembly to get results that were going to be a long shot to achieve by putting things together by hand.
The last part that has shortcomings is the base. In itself there wouldn't be any issues if everything else was perfect and the cat going for a morning jog was evenly dispersing their weight and force. In the real world, whenever a bump is introduced or greater left/right force there is flex in the base. When one side of the base flexes, so does the other. At the conclusion of whatever caused the base to flex, we have positive feedback as the base attempts to it's resting point.
What this base flexing leads to is yaw oscillation otherwise known as speed wobble. When the wobble gets bad enough it also introduces roll oscillation which is where some may have seen it come off the base, or as Chris and I experienced some drag and slowing of the wheel. Pretty much works the same way as speed wobble would occur on a skateboard and the remedy should have been adding two additional frame rails. Still the need for a dampener for yaw correction, but the two additional rails would have been an easy route to make things function a whole lot better.
An example,
Minor issue, the wheel is just louder than it needs to be. The track pad could be a little thicker and denser while there also needs something to deaden sound on the outside of the wheel.
So now what to do about the fixes. To keep the base from flexing, I'm just going to place a 1" x 12" board between the bases, 31.5" long. This could be enough to combat the speed wobble, although the ideal solution is to introduce a yaw dampener. For the sound dampening I think I'll just use some packing material and shrink-wrap then cover them with material so the cats don't un-stuff the wheel.
I think I'll have my Dad over soon to have a look so he can draw up something that will be complete overkill but spinning so true that it could be a primary part on a NASA space probe heading to Jupiter. That's how he rolls, and as happy as I am about the help he'll give me, I'm not planning on making any of our cats a space monkey.