Is there a way to get my organization's DHCP to assign my computer multiple IP addresses, then balance my traffic over all of them? (i.e., get a high downstream while not raising red flags)
The problem is the webserver that you are connecting to has no idea that IP1 and IP2 are both you.
Thus it is going to think that they are two different computers, and serve up the same content. What you are looking for is something like Bittorrent, but the current technology makes it rather difficult to request different parts of the webpages through different connections without downloading the whole webpage first.
In the end, it is possible from your side, but without support for load balancing on the other side, this is difficult.
Simple answer is no. You wouldn't be able to direct traffic based on destination using the same default gateway. If you have software that will let you choose the interface that is being used, you might be able to accomplish this. But for the most part, this isn't easily done without some other appliance or software to load balance the requests for you.