What to do about my trem?

[Sascha Franck]

The problem might be that what I want is at odds with physics. I rarely use the trem for big dive bombs, I want it to be sensitive for subtle vibrato. The kind of thing where I can just push up a little with a fingertip while playing and get a little vibrato.

But that kind of sensitivity is exactly what’s giving me problems when I’m digging in comping chords.

I gotta say I'm still picking quite hard and don't notice much of these issues, if any. Maybe check out dampening your springs, a little foam goes a long way and usually doesn't harm tuning stability. Also, I've got "silent" springs from HB on two of my guitars and they really reduce spring flutter almost completely.
Fwiw, I actually don't think it's got much to do with the vibrato system actually causing the issue. At least for me it's pretty much always been the springs.