I want to add this for anyone in the future that may come across this post.
I followed the advice of @Ramhound and the links he provided in the comments and created a new VM in Azure from the downloaded image of Build 14295, which was the closest available ISO build. I followed all the steps to connect my MS account to the VM for insider preivew. I changed my insider preview settings to Fast track. I even used my MS account to sign into the VM.
I checked the registry settings as suggested by this article and ensured everything there was correct. Even after all this, I still did not receive the 14316 build.
I diligently checked my machine multiple times per day. I manually checked for updates. I restarted the machine several times. I changed my settings to developer mode. Even after following all these steps, I still did not receive the build.
Finally! Today, after a week of waiting, I logged into the VM, and the build was there. I do not believe any of this extra work I did helped....but I did learn some stuff from this, so it was not a complete waste of time. I think I would have received the build if I would have just left my VM as it was and kept waiting.
tldr - From my experience, I think the best advice is to just be patient and wait. If you have waited a significant amount of time (2 weeks), then maybe look into the other solutions.