Can't help with an outfitter recommendation as I've never used any, but I've been in the Thorofare easily over 30 times in my life. You may want to consider that Cody based outfitters will pack into the Thorofare on the Deer Ck, Ishawooa, or Eagle Ck trails. Jackson based outfitters will likely pack in on the Turpin Meadows trail. Eagle Creek isn't a bad trail, but if any of your party is afraid of heights you'll want to avoid outfitters that pack in on Deer Creek or the Ishawooa as they both have hairy passes. (My neighbor lost a pack horse on Deer Ck just a week ago, and he's been in that way more times than me.) Turpin Meadows trail is long, but could damn near be considered a kids trail is so dang easy.
For the Thorofare being the most remote spot in the lower 48, it's gotten so crowded in the past few years that I've just stopped going in there. My idea of 'crowded' is probably different than yours, so don't let that sway you.....there's nothing like seeing that country on the back of a horse.