For those prices what you are going to end up with a a rifle very similar to a Christensen Mesa which is $1K less. I think to really get into a true custom with a reasonable weight you will be closer to 3200. I just put together a Bighorn, BnA trigger, Proof with brake, and a Mesa Precision stock. The Bighorn is nice because it is one of few that you can actually get shouldered prefits for and the only one you can get shouldered prefits from numerous vendors/smiths. For 2200 you will be getting a sub par stock and a barrel that is either heavy or a pencil barrel that will not handle shot strings. If that works for you then it is a viable option.
You do realize the stock you listed is on close out and there is one option? It is for a pencil barrel with m5 bottom metal(which adds money and weight). If you wait around to collect deals on parts and take a few years to piece it together you can get into the 2200 range if not that will be tough. Like I said you can get there if you compromise on components. The other thing is all you really get for that is maybe, and I emphasize maybe, a little better cycling action over a CA. I would still take the Fury over the build you are suggesting unless I was hell bent on not having a SAAMI chamber which I am not. I feel the OP has the info he needs. I am out. PM if you have any specific questions.