Just to go a little deeper . . .
No matter what size/number of springs you put in there, when you add tension (bending the string(s)) the spring will open.
More, or stiffer, springs will open less, but they WILL open, thus dropping the pitch of the other strings.
It's a balancing act, and what happens when you add more weight (in this case, the weight is the tension on the string) to one side of a fulcrum?
To restate; a Trem-setter, or Back-Stop (if you can find one) is one way to keep in tune and keep full range on your trem . . . the other is to set the trem up for no pull-up and keep the springs over-tightened so that you can bend and stay in tune.
You can also block the trem totally, but you don't seem to want to go that road anyway, do I won't mention it. ;-)