I tested, and am successfully using a solution to this problem. It starts with installing the DD-WRT 3rd-party firmware on a common wireless router, as suggested above by ace_omega. However, for my purposes, I configured my router as a client rather than a bridge or repeater.
Unfortunately, for you, I don't believe you will be able to install DD-WRT on your Westell 7501 router. The good news is that there are a great many routers that are compatible with DD-WRT and you should be able to find one selling (on eBay for example) for pocket change.
Here's what you need to do:
1 - Get yourself a router that appears on the following list:
2 - Avoid any that appear on this page:
3 - Read this:
4 - Go to DD-WRT's download page, and enter your router model in the white box.
5 - Download the DD-WRT firmware that has been approved for your device.
6 - Read the installation instructions that are specific to your device.
7 - Install DD-WRT. Reset router.
8 - Open the DD-WRT control panel in your browser (192.168.1.1).
9 - Within DD-WRT, go to "Status" > "Wireless."
10 - Click on "Site Survey."
11 - Click "Join" next to the wireless access point to which you wish to connect.
12 - When you see the message saying that you have successfully joined the network, click "Continue."
13 - The screen will automatically take you to "Wireless" > "Basic Settings" where you will see that DD-WRT has changed your wireless mode to "Client" and set your SSID to the same as the access point you have elected to join. Click "Apply Settings" to confirm these changes.
14 - Watch the information presented in the top-right of the DD-WRT control panel. Within a minute or so, a "WAN IP" should appear indicating that your router has successfully connected.
15 - You should now be able to connect through your router by plugging into one of its LAN ports.
I conducted my testing at a friend's apartment so I could look at things from the perspective of his router as well. I was able to successfully connect to his router by setting my router in bridge mode. I did not, however, need to make any sort of configuration changes to his router to enable successfully bridging to it. (In post #3 of this thread, you said that one needs to configure both sides to enable bridging.) My friend reset his router to factory defaults before we started just to be sure there was nothing that might confuse things.
While I was using bridging mode, all of my requests were forwarded on to the upstream router (as you said they would in post #6 of this thread). For example, my PC showed up on the upstream router's list of DHCP leases. Bridging should work for LieBot if he is only looking to connect one machine, or if the apartment complex in which he lives does not limit the number of connections (which is probably the case). However, it would not have worked for me as I was looking for a way to connect multiple machines through one upstream connection. Hence my need to use 'client' mode instead.
One reason I conducted my testing at a friends's apartment was so that I could be sure that I was not at any point allowing my wireless router to present itself as an access point anywhere on University grounds. After configuring my router as a client, I confirmed that it did not present itself as an access point, nor did it in any way effect the ability of people to connect to the upstream router using a variety of WiFi-enabled devices. After extensive testing, I am very certain that use of my client-configured router will not cause any problems for any of my University neighbors. But, thanks for your input !