September is a wonderful time to climb in the Cascades, actually it is my favorite month! The only issue is that the glaciers will be more broken up and will require some route finding, but the standard routes on Baker, Shuksan, and Glacier should still be fine. (The other peaks you mentioned do not involve glacier travel.)
I have climbed with a number of folks who got their start by taking an AAI course, any they spoke very highly of the guide service and their experience. I don't think you can go wrong with AAI.