Don't carry spare inner tubes, just a patch kit. You can pick up a few bike-specific multi-tools at a shop (usually hex wrenches, or regular wrenches of various sizes). You can either carry a pump or a CO2 cartridge-type pump (uses the same CO2 as a BB gun). It DOES suck to push a lame bike.

Good bike lock (duh). Bike gloves: the handles will be HOT, and it helps decrease vibration (and hand numbness after a while). Water (probably a camel back) since it'll be HOT. Make sure you've got good reflectors, or better yet, lights. I've almost gotten killed more times than I want to admit. A good bike helmet.

If possible, why not just take some clothes to the worksite at the beginning of the week, and just pick up them up on the weekend? Saves you carrying clothes.

Personally, I'd drop anything EDC that isn't used at least once weekly. For me, that means I wouldn't carry a multitool, PSK, FAK, or the like. Wallet, cellphone, spare cash. Maybe a flashlight, if you ride home after dark.