I found out some additional information:
For the USA proxy I was using, Netflix did not recognize it as a VPN/proxy and gave me the full USA version of Netflix. StreamFab worked fine.
For the UK proxy, Netflix recognized it as a VPN/proxy. When this happens, instead of blocking itself entirely, it instead gives a "geo restricted version" of Netflix after log-in. In the "geo restricted version", it only shows Netflix original content, and content that it has licensed to be viewable world wide. Anything that is restricted to certain regions will be hidden.
It is this "geo restricted version" that StreamFab is not able to download from. After clicking any movie or series, StreamFab shows a blank black page.
There is a quick way to tell if you're logging in to the "geo restricted version" of Netflix:
Pick a custom profile icon in Netflix that is different from the default. You can find it under "Manage Profiles," click your profile, then click the little edit icon next to the profile logo.
When logging in to the "full" version of Netflix for your country, you will see the custom profile icon.
When logging in to the "geo restricted version" of Netflix (VPN/Proxy detected), it will show the default profile icon instead of the custom one you selected.
So, again, to summarize:
StreamFAB does not work the "geo restricted version" of Netflix.