Can I choose which internet connection the browser uses?

I have 2 connections, one ethernet one wifi. The wired goes through a corporate LAN, the wifi to a router with a DSL connection. I'm trying to get to a site that the LAN blocks (it's not inappropriate, but they consider it wasted bandwidth), but the browser always uses the wired connection.

Is there a way I can tell the browser which connection to use for that site? Or can I tell Firefox to always use the wifi connection and let Internet Explorer use the wired?

Note 1: In case that wasn't clear these are 2 different connections to the internet, not 2 paths to the same router.

Note 2: The LAN doesn't exactly block the site, it redirects it to an explanation page. The browser might be smart enough to try the other connection after no response, but a redirect isn't no response.