https://www.thespringstore.com/pc125-703-6250-mw-1090-cg-n-in.html
This spring still wasn't perfect but much closer to holding it down completely and much more stable overall. Now it at least returns to the same spot each time. And I think I actually like it floating just that little bit.
As far as how to install it, I did not get any special pliers or anything. I just unscrewed the side screws that hold the top and base together / where the bridge actually rotates. Then it was easy to put the spring in place and screw it back together. Way easier than trying to squeeze a quality spring into that spot.
Still much more work to make this guitar actually nice but I'd say it's at least giggable now.