Well, by my NatGeo map, if my math is right, you're looking at making about 18-20 miles in 6.5 planned hiking hours on Day One. That's a pace of 2.75-3mph including any stops you make.
Elevation gain is ~5700', with about half that during the first 4-5 miles and about half during the last 3-4, with 10 relatively flat miles in between.
I usually average 3-3.5mph in the Sierra over that kind of terrain including stopping for views, refilling water, etc, but that's me.
Also, I'm going to be in Yosemite/Ansel Adams Wilderness the last week of September and sunset is in the 6:50pm to 6:45pm ranger during that time, which means you'll be walking the last 3-4 miles of your trip in failing light to darkness.
As Bob said, Illiloutte Creek is the only water source between Happy Isles and Lower Merced Pass Lake, which is a distance of 16.5 miles, and late in the season it may not be reliable for the entire distance, so keep an eye on it. I like to have about 15-20 oz of water per hour, so 2 qts for me would last 3-4hrs. At my pace, I could probably make 16.5 miles okay on that knowing I could take up at the end, so long as the heat wasn't excessive, which it shouldn't be at the end of September.
What pace are you capable/comfortable with? Are you okay hiking and setting up camp in the dark?
I believe there are showers at Curry Village for pay, and I'm pretty sure anyone can use them as long as you are willing to pay.
There will be bear lockers at the TH, or plenty of other places. You can use those for storage. Not for valuables, but I've left clothes in ther for 4 days with no problem.
Human waste that can't be buried should be packed out.
Edit: Also, since you are showing up at Noon, did you note that permits are only held until 10am the day of and then cancelled, unless you call to say you will be late? I don't know if you have to call the day of to say you'll be late, or if you can call they day before. Wouldn't want to show up to find my permit given away.