When I had guitars with a Floyd, I did the 'suitcase handle'. Just gotta remember that the lighter the string, the less tension you have before breakage. I was told by a guitar tech that on the road, you have to the 'suitcase' pull because you don't have time not to.

However, to the initial problem: Remember that a Floyd Rose is a floating tremolo. That means the the tension from the strings must equal the tension of the springs.

If you stretch the strings enough and it still flats out, you may want to bring it in for a set up. Not cheap but worth it if your springs need an adjustment or replacement.