I don't know if this will help or not as I've never touched a Lynx, but this was my experience SBS'ing a Saiga 12. I am far from claiming any kind of expertise and hopefully someone much more knowledgeable will come along with better information.
I can only assume that the gas system in the Lynx is similar to that of the Saiga. The barrel on my S-12 is right at 11.25" as I was trying to obtain a certain look for the finished shotgun. I was using a conversion kit from Dinzag Arms that allows you to use standard AK handguards. The kit included the front site post (uses set screws to anchor it) and the AK74-looking muzzle brake. To use the muzzle brake I had to re-thread the outside of the barrel which I did myself.
I already had another S-12 which was picky with different power loads with a full length barrel so when I started working on the SBS I knew that I would have to modify the gas ports as the barrel got shorter in order to have enough gas to cycle properly depending on what loads I wanted to use. My goal was to be able to cycle high power loads and also lighter low power bird shot loads. I chopped the barrel in small'ish increments and test fired different loads each time. If the lower powered loads wouldn't cycle, I would press off the gas block and open up gas ports a little at a time until those loads would cycle. My gun was a four port barrel from the factory (see example in the photos below - those are not photos of my actual barrel). Before I was done I had added a fifth port. I can't find my notes, but I think I first chopped it to about 15" and then a couple more increments until it got to where it is now at 11.25". Right now it won't cycle the very light loads, but I could probably fix that by opening up a couple of the ports just a little more. Keep in mind that the adjustable gas plug (at least on the Saiga's) just covers or uncovers the gas ports in the barrel in order to regulate the full power available via the ports. Once the plug is open, it's open. If there's not enough gas to cycle loads, you'll have to allow more gas to come through the barrel by enlarging the ports or adding ports if multiple smaller ports are used like on the Saiga.
Assuming that the Lynx has similar gas ports to the Saiga, I would be pretty confident that you will have to modify the gas system a bit to get it to cycle properly at 12 or 12.5" - unless the gun was way over gassed to begin with.
My gun showing front end with and without the brake.
Example of Saiga gas ports (not my gun)