It should be able to make that 1/4 turn. What I've done with that type of spring (hood support) is grab that tang with a pair of vice grips to give you more leverage to wind it up and get the tang to it's proper position.
If you can't get it done, send it back. Improper fit is pretty common with repop parts. It does look a little thicker than the original. You might be able to unwind the original spring enough to take care of the door handle sag.
I thought that the door handle sag repair spring required partially disassembling the latch and putting it around the square hole where the outside handle inserts. I bought those, but decided not to install them to avoid dismantling the door latch. My door handles don't droop enough to worry about.
