Will installing a game on Xbox allow me to bypass a location at which the disc becomes unreadable?

Once installed from the disc to the hard drive, the Xbox only uses the disc at startup in order to verify that the correct disc is in the drive. Past that point, it won't read the disc again (typically). I'm a little surprised that the installation passed the bad area of the disc, but the game doesn't.

What you might want to do is contact customer support for the game, usually there's a phone number, email address, or website in the paper manual that came with the disc. If it was defective when you opened it, and you can't return it to the store, they might be willing to exchange your copy somehow. Since you live in a different area than where you purchased it, they might give you a hard time, but it's at least worth a shot. You'll want to keep your receipt and all the packaging, as sometimes they'll want you to prove you bought it by mailing it to them, or sending them scans or pictures of some part of it.

If that doesn't work, as a last resort you might want to take the disc somewhere locally that sells or rents used discs (DVDs or video games) as they generally have resurfacing machines that you can use for a nominal cost. Doing this may or may not make any improvement in the status of the disc, but it's still worth a shot if you run out of options.

When a game is installed to the hard drive/USB stick the Xbox 360 will read from it exclusively after verifying the disc in the DVD drive is the correct and valid game disc.

If the disc is scratched or otherwise damaged you probably won't be able to install from it.

This happened to me with Lost Odyssey disc 3. There was a certain point the game that I could not progress past the train scene. Due to the scratch I could not play past the scene and I got read errors when trying to install the disc. Microsoft wanted $20 USD to replace just 1 disc! What I ended up doing was renting the game and installing from the rental, then returning the rental and using my disc to authenticate and play the game.