I could take this challenge and come in at ~$325 in Washington DC.

Budget:
- Light ($75)

Amazon.com: Nite Rider Lumina 550 Bike Head Light: Sports & Outdoors

- Cables and Brake Pads ($25)
- 7-speed Chain ($10)
- Tires ($25)
- Rear light ($10)
- Used, rigid-fork, steel MTB from the 1990s ($180)

Bikes are numerous, if you know how to search: Nishiki, Fuji, Giant, Univega, and Trek (heck, even Schwinn) sold steel famed MTBs in the very late 80s and early 90s.

These will be 3 x 7 speed, with gripshipft or maybe lower end Shimano thumb shifts, cantilever brakes, quick release hubs. look for ~20 inch frame size. It can be done.

If you need it all in 1 week, you'd best order the light first, since that's mail order...