I finally had a chance to try a few of the suggested settings for addressing the floatiness and centering issues. Specifically, I played around with the amount of cockpit camera movement, the audio settings to try to make the tire scrubbing more audible, and FFB settings to try to reduce the aggressive (over)centering.
I think they all helped a little, but it gets tough after a while to know how much it's the settings making a difference, just getting used to it, or placebo. I wouldn't say any of them resolved it fully enough to be certain it made a meaningful difference. I was able to get the scrubbing loud enough to hear more frequently, but the caveats are that (1) it is still often hard to hear, (2) some cars are harder to hear it on than others, and (3) doing so means that the engine is less audible, more so on some cars than others. So, I think it is an improvement, but not ideal.
I'm not sure where I stand on the cockpit camera movement. On the one hand, more movement does seem to reduce the floatyness a little, but it creates its own distractions as it doesn't seem like natural head movement. I'll have to play with that some more.
As for FFB, I have found several types of tweaks that tame the centering, but all are essentially means of blunting the FFB altogether. First, you can turn the various settings down so that the centering is weak enough that it isn't so disruptive and less likely to overshoot the mark and lead to oscillation. But, I find the level required for this to be quite weak and too muted. Second, you can crank things up, essentially clipping your wheel, which by necessity reduces the extent to which spikes such as aggressive centering are felt. But this obviously makes the FFB much less informative, and can have a host of other drawbacks (FFB as an on/off sensation rather than a matter of degree, tiring, etc.). I think this is essentially what some of the suggested extreme vertical load and/or lateral load settings (e.g., 400%)accomplish.
A third approach that I've settled on for now is using a little bit of Speed Sensitivity Steering. I know that you generally want to avoid that when using a wheel, but it seems to help here. It essentially dulls the FFB a little bit at high speeds, which helps tame the centering. Not ideal, as it dulls everything, but it seems like a decent compromise for now. I haven't settle on exact settings, but for now have it at 20%, kicking in at 30, and maxing out at 250. So, it is a relatively subtle effect, and I can still feel the aggressive centering but it is less disruptive and less likely to overshoot the mark.
I am encouraged by seeing that S3 plans to add some additional FFB parameters, including centering, in a future update. Hopefully that will provide a more optimal solution, but this seems adequate until then.