I've never had to deal with it myself but when the ranger does a headcount at check in and sees you've got too many people, you may not get to launch or it may be time to boot some of your participants off the trip. A lot of volunteer rangers (like myself, Rich P, Mongo, Buck, etc.) are on MB and someone who's dealt with it, or the full time rangers could answer your question better. Send a PM to MB user "BLM Moab" with a link to this thread and the head Ranger for Westy may answer it for you.
If you don't want to stuff people into dryboxes for a few hours while you rig until you get around the first bend, just tell the extras that's what they will have to do and they'll probably pass on the trip anyway.
But seriously, you could also just put up a message here and other groups (Utahrafters, GCPBA, etc.) and see if any other permit holders are willing to cancel excess spots on their permits so you can enlarge your permit.