I had posted about this a long time ago in another thread. The spring goes around the stock lamp spring, not the tailcap spring. The purpose of the spring is that the Pila spring has a large inside diameter that will accomodate the larger diameter Pila batts. The problem is that the Pila batts are sufficiently wider than 123's to prevent the Surefire LA spring from coming into contact with the body of the light. All the Pila spring wrapped around the LA spring does it that it bridges the gap allowing contact.
All you're doing by putting the spring in the tail cap is furhter compressing the LA cathode spring.
Look, try this and see if it works, as a trouble shooting measure. Feed the batts into the light through the front end with the tail cap on. Put in the first battery. Take the second battery and work it into the spring of the LA. It can be coaxed in. The batt will them be sort of stuck in the spring. Now just screw the bezel onto the body with the batt stuck in the spring. This should allow the cathode spring to come into contact with the body and you should have light.
BTW, LA = lamp assembly