Yup, chucking on more springs and then tightening up the claw will help, but floyds are a pita in general, which is why my new axe is a hard-tail. You use 11's, which is always going to be harder on the trem during bends etc, so the only partial solution is to add more heavy duty springs to the claw and crank it a bit back into the body. You'll have to work harder to use the arm, but apart from using the tremel-no to stop it completely there aren't many other solutions.
Let us know if the doohickey you ordered works, though! Very interested to find out more solutions for floating trems! :)