Try blocking the trem... you said you're a 'hard hitter' and while I'm not a hard hitter, I find that in the lower positions if I play hard it causes the bridge to flutter a bit.
When you have the trem blocked, if you still get the sound, really stuff something thick between the springs while the trem is blocked (that way you can keep good tuning while really making sure the springs are dampened.)
If you STILL hear this phenomena, then you know it's something else resonating.
As usual, check that everything is tight before you even try this stuff. I had a loose strap button vibrating on a guitar and could hear it whenever played without a strap, it added a strange buzz to certain notes/chords ... had me buggy for a few minutes when I realized that it went away with the guitar hanging from a strap... so it's all relevant.
As far as simply being an intonation problem, it well could be. Not that your guitar isn't 'properly' set up, but it's wire and wood, and no guitar is ever perfectly intonated. Some will show this kind of thing, others, even though they aren't 'perfectly' intonated may not show it enough that you notice.
I had a customer bring me a CHEAP CHEAP CHEAP, Harmony that he bought at Best (not Best Buy, BEST... this was a long time ago) and couldn't figure out why some many of the chords were out of tune even though the guitar was in tune. Twelfth fret, nineteenth fret, all over the board, were out. No amount of tuning, leveling... intonating, could fix this guitar. (Of course, it was a whopping $90)
That's an extreme case, but it serves my point.
Anyway, long reply, but I think I covered everything I could think of.