So isn't that the blade gap being different? Are blade gap and clamp down connected in some way? I don't know much about the clamp and blade chatter but blade gap is something I understand. You're suggesting to get a razor with smaller blade gap?
No, I'm not suggesting a razor with a smaller gap. The two design styles don't really have anything to do with blade gap. You can use the Piccolo style of a well clamped blade and still achieve whatever gap you want. Timeless razors clamp very well, like the Piccolo, and offer .68 and .95 gaps. I didn't mention them at first because it's a more expensive razor and you indicated you weren't looking for that right now.
So, less gap might help, but that's not what I'm suggesting. I'm suggesting you might try a razor design that clamps the blade near the cutting edge, but that doesn't mean it has to have very little blade gap. Here's a picture of the Timeless and a Rockwell that might better show how you can have a base plate and cap that clamp well by design and still have whatever gap the maker chooses.
This isn't my picture, but you can see that clamping really depends on how the base plate is designed without having anything to do with blade gap. The Timeless base plate on the left extends all the way to very close to the cutting edge. While the Rockwell on the left only uses those two little posts to support the blade and they aren't near the cutting edge.
That's not to say that the Rockwell/L55 design is bad. It's not and many people here love razors like that. I've just found that the Timeless/Piccolo style works better for me and my whisker characteristics.
Hope that helps.