There isn’t a lot of spring for pushing, and that insert will really get in the way. So while I don’t know if that would hurt the spring, it probably won’t be very helpful for your installation.
If you’re having trouble getting the larger base coils started, you may simply have to grab some fine needle-nose pliers or forceps, hold the spring 180 degrees away from the end of the bottom coil, and sort of “lean” that end into the recess. Once it’s there, start turning the whole thing in the direction the spring is coiled until you get all of the bottom turn in the recess.
After that, you can use your tool (no longer holding the spring wire) to push that bottom coil all the way into the recess. This may take a combination of patience, LOTS of light, and maybe some magnification.
This is a big part of the reason the TM says not to take out the extractor spring if you don’t need to. Armorers really know how to cuss; maybe this is why.