Puck Daddy just put up an interesting post detailing the best shootout specialists of the last couple years. (Apparently, Ales Kotalik and Phil Kessel are your winners)
The post ended with the line: "Now if only some of that mattered in my fantasy league ..."
My suggestion(s) are simple.
1. Don't let it affect goalies
Goalie stats are fine the way they are. The shootout IS, after all, a gimmick. And it's the hardest on goalies so let's leave it where it is: already huge since it decides numerous goalie wins.
2. GDG + GWG = the new GWG category
Some people don't like GWG as a category, but it's never really bothered me. Not when there are bigger evils like SHG/SHP (totally effin' random and almost never happen relative to relevant stats like goals or +/-).
By adding those GDG that Puck Daddy mentioned (it's not like it would be/is a hard stat to track) you'd guarantee a GWG stat for every game and it would add an interesting wrinkle of strategy. More frequent GWGs means the stat category would be less random and annoying.
3. Add a shootout goal category (better yet, make it a fraction)
A lot of fantasy sports leagues make certain stats more important than others or use other triggers. What if a shootout goal (which, let's face it, in the scheme of a game is pretty damn important) counted as half of a normal goal? It wouldn't be hard to track and would give fans one more reason to jump out of their seats.
If the shootout's going to stay we might as well enjoy it. And what better way to enjoy a gimmicky event than to connect it to fictional competition?