I'm pretty new to the 300 whisper also, but you may really want to consider a barrel that has a pistol gas port location. This would allow you to go much shorter. I had one cut to 9" from a 16" barrel and it functions fine without opening the gas port up. I only shoot subsonics so I didn't get an adjustable gas block. I also use 220gr loads and have had luck with H110 at 9.2grs, it gets about 1025fps out of my barrel.
If you really want the carbine gas system, I think you could do it, just a little tricky when you get really short. If you went as short as 10.5" you could probably be ok with a little work. You most likely would have to drill out the gas port to make it function though. Maybe a port opening of .100, but I would start at the factory size and work up.
The suppressor will help the gas system function by adding some back pressure, so you got that going for you too.
When you use the carbine gas system, the distance between the end of the barrel and the gas port is really short and doesn't allow much room for the gas to do it's work.
This is a photo showing a 9" barrel with the pistol gas port location.