Several things. First is that the arm is a lever. A lever gives more leverage. So using our hand instead will produce less leverage, so less chance of breakage.
Second is with the bridge like that, there's lots of movement to raise the strings' pitch, but little to drop it. The bridge needs to be able to move both ways, so tighten the screws underneath until ...
To get the Jeff Beck floating bridge thing, you need to be able to go both up and down, and I think you've used up all the down. Here's Carl Verheyen explaining how to set up a Strat so it floats, which should guide you.
I would tighten the claw and retune to balance. That should make it behave more nicely and allow intonation. Best of luck.