After a long wait, the transfer kit finally arrived. And bizarrely it worked fine.
One end of it hooked up to the new hard drive while the other end plugged into a rear USB port on the console with the old hard drive installed. Because of the amount of stuff I had on my drive, the process took about an hour and a half. Weirdly enough, all the crap I deleted from the new drive magically appeared again after the transfer so I had to get rid of that all over again. Oh well, if that was the worst thing to happen, I got off pretty light. It feels pretty awesome to have all this extra space now.
Pretty much this kind of awesome.