As it comes out of the box: no, the full-auto Ciener kit cannot work with a Lightning Link (LL).
I thought maybe you could modify the Ciener kit, but the more I tried to prove it, the less convinced I became.
Here's a pic of an M16 bolt carrier and AR15 bolt carrier (ignore the PC BC on the bottom).
[img]http://www.ar15.com/content/docs/AR15-M16Parts/boltCarrier.gif[/img]
As you can see, the place where an M16 BC trips the auto-sear is 1.5 inches in front of where an AR15 BC trips the LL.
Now, here's the Ciener kit:
[img]http://www.22lrconversions.com/m16-prt.gif[/img]
Looking at the Ciener kit, the "bolt carrier" actually stays in place and the smaller .22 bolt slides within it. I think the full-auto adapter piece is somehow clamped to the underside of the bolt (behind it) so that when the bolt returns forward it trips the auto sear. Because the whole kit stays in place and only the bolt moves, I don't think it's possible for the auto adapter to be modified enough to allow it to clear the LL, yet trip it when the bolt returns forward. When the bolt tries to go backwards, it's either gonna hit the back of the "bolt carrier" or at best, it's still gonna smack right into the buffer which keeps the kit in place. Either way, not enough room to move.
At a minimum, thanks for the question cause it made me go learn something new [:)]
[:I] LL [:I]
PS: Don't be fooled by the handle. Sadly, I don't have an actual Lightning Link, I just play one on the internet.