Rockhound - you're installing a CAR stock, which is NOT the same as a buffer tube! Assumptions like that can get you in trouble! Note that your CAR stock has a locking ring. That ring secures everything, unlike the standard buffer tube the CAR stock is not intended to be screwed all the way in! You have some play there so you will be able to make sure the stock is vertical with respect to the rest of the gun.
Screw the stock into the lower just far enough so that it will keep the buffer pin from launching itself (seems you've covered that one already). Notice that your stock has a locking ring? Well, get the pivot pin detent and spring in place, hold the cover plate over that with one hand so you don't mangle the spring and tighten the locking ring down good and snug, and you're done. Shove the buffer and spring in there, and it should stay put unless you've screwed the stock in too far and it keeps the detent from protruding far enough to do its job.
Oh, and make sure the stock is aligned with the rest of the gun before you tighten the locking ring down. If your BC still hits something when you close the receiver, make sure your bolt is fully extended from the carrier or it won't engage the barrel lug and it will protrude rearward enough not to clear. A simple thing, but can fool you if you're too focused on the other problem.
Hope that helps you.