Thanks for the links. Made me think of something else. The Weber has an electric choke, and I found the colored wire (red/silver?) that went to the choke on the Mikuni. I followed the wire to the harness, and stripped the end of the Weber wire and just shoved it into the female end of the harness where the red/silver wire was. I didn't check this point with a test light, to make sure it was getting power when the key was turned on. That might be my problem, in that the Weber electric choke isn't working.

So I'll break out my test light tomorrow, and make sure that connection is getting power when the key is turned on. If not, I'll find one in the harness that does, hook it up there, and see if the rattling noise goes away.