We went up to Rainier twice in Aug of 2011 and 2012, it was warmer than in Jul I guess and there were 3 of us. Here is the gear list anyways, maybe you'll find something useful. I would keep storm pants and warm bottoms, but leave the softshell pants at home. You don't have a good warm puffy it seems, so two lighter probably make sense..
Individual checklist:
1. Warm cloths (it will be very cold on the top, make sure you have very good gloves system: I used three pairs last year: light liners for hiking, warm mountaineering/skiing gloves above and a mittens hardshell above, make sure you have really warm socks with liners)
2. Ice axe with a long sling securely attached to the harness
3. Crampons
4. Harness
5. Helmet
6. Three prusiks (5mm)
7. Belay device
8. At least 3 more locking biners (one for ATC, 1 for prusik, 1 just in case)
9. Couple of slings attached to biners
10. Sleeping bag
11. Sleeping pad (inflatable termarest works better if we'll have to sleep on the ice)
12. Head lamp and additional set of batteries
13. Food for good dinner and light breakfast: avoid anything fried, fatty, any kind of beans, fruits or vegetables (potatoes are ok ) .
Prefer carbs with low glycemic index (pasta, rice - lots of freeze dried options available).
14. Water bottles or camelbak (camelbak is better to avoid frequent stops, bottles are still very useful to cool down the water before filling camelbak)
15. Snacks to have on the way (gels, cliff shot blocks, energy bars etc) that can be eaten without stops
16. Spoon
17. Hiking poles
18. Lipstick with high SPF rate
19. Glacier sunglasses
20. Ear plugs (sleeping at the camp Muir/Shurman can be impossible during high season)
Group (weight to be divided between all members):
1. 2 jet boils, we will need to melt snow
2. Fuel for jet boil (two NEW medium/small size jars)
3. Bleach drops to purify water
4. 30 meters glacier rope
5. One tent
6. GPS and additional set of batteries
7. Cell phone
8. Medical kit (climbing tape, painkillers)
9. Knife
10. Toilet paper
11. Goggles
12. Tea bags
13. Two pickets with sling and two biners (at least one must be locking)
14. Ice screws
15. Sunscreen with high SPF rate
27. Pulleys
20. Tiblocks