"Donnie, you're out of your element." - Walter Sobcezk
Allow me to kind of retract an earlier statement; there is a right way and a wrong way to install a vintage trem, naturally. The six anchor screws are there to hold the trem in position. That's it. It has more in common with the neck plate than any other component on the guitar in that it connects two components to the major system. It is not a component of intonation, string height, or action. The only other function it has is tilt the entire system forward against spring tension. It is by design a static system with the manual capacity to disrupt a static system and return to the resting state.
If you look at the bottom of the trem plate you'll see that it is not flat, it has a knife edge bevel on it's leading edge that extends to the pivot point of the system, the screws, to allow it to travel along a single axis. (further back on better after market plates) This allows the system to pivot forward and slack the strings. The sprung energy returns the system to its resting position.
The proper way to "set-up" the trem plate it to set the screws down far enough so that the rear edge of the plate raises above the body then tighten down the two outside anchor screws only enough so the plate rests on the body. The four center screws are then tightened down until the heads almost touch the plate. The plate should not slide on the screws when the trem is engaged at all. That is bad. The trem should return to pitch once dis engaged. If not, the spring and claw need to be adjusted. This is not "floating" system so you can't bend it sharp and expect it to return to proper pitch, sorry. I Stated before that there is no magical way to set the trem, what I meant was; the person that installed the trem on your guitar (and 300 other guitars that day) did not spend that much time installing the bridge, not in China, not in Corona, California, maybe at the custom shop, maybe. But even they are working with an imperfect system designed 65 years ago.
@rparker There is nothing to be afraid of, you are not going to ruin anything by changing the settings even if it isn't correct, it just wont function properly. It actually requires better under standing of the system to twist a tuning peg than tighten and loosen these screws. If you installed locking tuners on your guitar than you've already undertaken a much more difficult and sophisticated task. That said, Installing locking tuners may have caused the problem you are experiencing. You are locking the strings into position at the head stock but the rest of the system is still compensating for sag and fatigue from the tension of the strings. Now the load, in theory, is being transferred to the bridge end of the strings instead of over the entire length, resulting in more regular maintenance at most. But that's only a theory based on experience and some deductive reasoning.
Also, If removing the back plate causes the problem to become louder (acoustically of coarse) then, logically the problem is originating somewhere in the spring cavity. In that case everything I've written above is superfluous.
That is my two cents based on actual experience. If you're still reading, here is my rant:
If you have no experience with a subject and you state so in a post it is good form to follow the statement with a question regarding the subject when your statement is in contrast to a previous post, not your opinion of the subject that you have already professed to have no experience with. It shows respect to the person who has offered his knowledge and expertise on a subject. So instead of stating, for instance "I have never tried what you are talking about but I think you are possibly an idiot" Try instead, "Would, adding a grommet between the plate and the body cause any loss of tone or sustain?" The answer would still be; No the addition of the grommets do not interfere with the contact point and axis between the bridge and the body.