that should be a great time to go. There are plenty of excellent, graded tent sites below "camp hazard", maybe 15 minutes below the drop to the ice chute. Potential flowing water too. Bring a 2nd tool for the leader!
I recommend going light and carrying over--thus avoiding downclimbing/rapping the ice and getting to see another side of the mountain. You mentioned being a Rainier first timer, but I figure you have some glacier and ice and steep snow experience to be considering this route...right?
Here's a photo of the exposure to be expected on the ice chute as well as a link to my cc.com trip report last August:
kautz report aug 2004