Ya he's right, do the whole thing.
That said, a half a trip is still better than no trip. If you have a choice, opt for the second half as it is a few days longer. Either half provides the GC experience. Joining a trip at Phantom is really getting thrown in the fire, as the first 4 rapids you run are 4 of the classics and there are 2-3 solid days of really fun boating thru the heart of the canyon. The group also has a solid group dynamic going that will take a few days to assimilate into. Hiking out at Phantom just plain sucks cuz you are leaving your tribe, and that hike up in July is a mother. Start your hike as early as possible either way. Indian Gardens has good shade and water, so you will get a respite around the half way point of the hike.
You should be able to work out a shuttle between the incoming and outgoing hikers, that way your vehicle is only parked for a few hours at GC Village, and all vehicles can end up in flagstaff or wherever the trip is based for convenience. If you hike out you will be ready to get the heck out of GC Village.
South rim has more and better access, and way shorter drive time. Getting cars to the north rim can add a lot of drive time, but it could be convenient depending on where you and the other hikers live. There are numerous drainages you can access the south rim, but Phantom is the most convenient and popular from my experience.
You will now receive a hundred different takes on all this as well as plenty of other advice, solicited or otherwise. Boaters who have ran the GC hold it in higher regard than any other river trip, and love to talk about it.
In any case, whatever you do, go on that trip!