I think I would recommend Rockwood Lodge as outfitter.  Mike and Lin are the owners there and they have a good outfitting business, cabins, etc. and also a nice bunkhouse where you could stay the night before if that interests you.  A good safe place to park your car, or I am sure they would shuttle a car for you.  They are close to the entry point on Lizz, and should have any information you need, and any gear you wish to rent.

We are staying in one of their cabins this August for a week with our granddaughter and did so last year as well.  We haven't used an outfitter in many, many years, and never for an entirely outfitted trip, but I would have confidence in recommending Mike and Lin. 

You can also arrange to pick up your permit at the outfitter if that is more convenient for you.  Didn't know if you knew that or not.