Minute 10:50 of the video shows the part on question. The white plastic piece has threads to hold the metal bar. If your metal bar keeps slipping, it means you may have worn threads and need to white replace the plastic part.

 

Yes, I'm sure that would solve the problem, acg. But I don't like that idea for some reason. I just got back from Ace Hardware and bought myself a couple of 10/32" locking nuts. Well, I say bought, but they actually gave them to me since they were only $0.28 including tax, which was very nice of them. I threaded the nut all the way up to the top of the bar and latched the white plastic clip on just under it. When you pull the handle, it pushes the bar down, which pushes the plastic clip down and actuates the latch, so I only needed one nut on top of the clip. The other one is ready in case the same thing happens on the passenger side. The handle works again, and time will tell if it will keep working. There's no way that it can slip past the nut. The only failure mode I can see here is if the clip somehow comes all the way open and completely falls off the bar, but I haven't observed that to occur yet.

If you want to do this yourself, definitely get the locking nut because the bar is not a standard size, metric or imperial. The plastic locking material allows it to stay secure, whereas a normal metal nut would have the potential to work its way up above the threads and stop working for us.