Welcome to the forum Peter and thanks for the intro!
I’ve never heard of a tangled mainspring described as a “hair ball” before! 
That’s a good one!
I believe that all the newer white alloy mainsprings are pre-lubricated with teflon and can be installed right into the barrel. I usually put a couple of tiny dots of grease on top of the spring before pressing the cap on. This probably isn’t necessary but for some reason I feel comfortable doing this.
Hopefully the spring will fit into the barrel while still in the retaining ring. If so then all you need to do is place the spring into the barrel with the ring resting on top and then just push the spring home. This works well for most mainsprings with normal folded back bridle ends but for hole end mainsprings that need to catch a hook on the inner barrel wall such as found in many pocket watches care must be taken not to damage or shear off the hook.
If you do need to remove the spring from the retainer so that you can insert it into the barrel with a winder then it can be removed from the retainer by working the inner coil out the same way that a spring is removed from a mainspring barrel. Another way is to place it over a large diameter winder barrel and then push the spring out of the retainer directly into the winding barrel. The spring can then be pushed out of the winder onto a piece of clean wax paper (or any other clean surface) by using the winders plunger.
This is all easier than I’ve managed to make it sound so if I’ve confused you in any way or if you have more questions please let me know.
I hope it helps though.
Enjoy Peter and again Welcome!
Bob