Took my car out for an extensive testing session last night. Switched to the TPMS readout so I could monitor temp by pressure changes. Originally my tires were 33psi cold, but it's colder now so they started at 32psi last night. Flat spotting was pretty bad and took 4-5 miles to disappear. The tires were 2, maybe 3 psi higher when warm.
There is still an imbalance with my car but I was really happy to find it moved to the front. Not really happy that I still have a problem, but instead that it's not with some other component of the vehicle. Will be much easier to get fixed once and for all.
My theory is this: The dealer road force balanced the tires when they were still a little square which caused weight to be added when it shouldn't have been. I'll bet if I coordinate with the service manager I can drive in and have them lift the vehicle ASAP, when the tires are still warm. Then they can remove them and balance them while perfectly round. I didn't feel them on my test drive with the other car (switched to have my tires) because the tires were already quite warm, and the prior balancing meant the imbalance is barely perceptible. Especially when moved to the front since it has more weight than the rear.
Which should leave flatspotting as my only issue. I tried adding 3psi to the tires and it helps with both the flatspotting and the other imbalance (makes it even less noticeable.) I don't really want to run my tires so high it will wear out the center of the tires.
The flatspotting is almost certainly due to the nylon used. Hankook's description of the tire:
The tire’s internal structure includes twin steel belts reinforced by a double layer of spirally wrapped nylon to blend handling, high-speed durability and uniform ride quality.
TireRack has an informative article on flatspotting with tires that use nylon:
http://blog.tirerack.com/blog/roll-on/flatspotting-try-saying-no-to-nylon-and-yes-to-polyamide
TL;DR version: tires that use nylon instead of polyamide cap have a "memory" and if you want to fix it, get another tire.
TireRack's solution is, of course, to buy new tires. Personally I think that's fine if I had researched the tire and purchased it knowing about that behavior characteristic, but not acceptable for a $50k luxury sedan bought off the showroom floor.
I'm going to call Hyundai and see what can be done. Wish me luck, I think I'll need it as I suspect there's nothing they'll do.