[todor]

Update:
I succeeded in connecting my BB to the internet through a computer, but only when the BB is enterprise-activated with a BES. It did not work with a Unite-activated phone.

The procedure is very simple: just plug in your phone via USB (may also work over Bluetooth - I haven't tried that) and start Desktop Manager on the PC. Enter your password and make sure the Desktop Manager is properly connected to the "messaging system" (which means that when you installed DM, you should have selected BES as the messaging system). That's it. The browser should now work even if you turn off your wireless and Wifi on the phone. Other apps may or may not work because it's only BES traffic that gets relayed (according to the documentation at least), and apps are free to decide whether to use a BES connection or to use a straight-up TCP connection that just goes through the carrier.

Thanks to everyone who contributed here.