"bash" command runs scripts in version 3.2, not 4.4
The error here is that the first path in PATH was to the bash executable, instead of to the directory that contains the bash executable. For readers with similar problems: double check to make sure any path you're adding to your PATH variable resolves to a directory, not to a file.
Problem:
I had added the following to my .bash_profile:
PATH=/usr/local/bin/bash:$PATH
/usr/local/bin/bash is not a directory. It's a file (well, a symbolic link to a file). So, regardless of placing it at the very beginning of PATH, the command bash will not find an executable file named bash inside any directory in PATH until it gets to /bin.
Solution:
In .bash_profile, setting
PATH=/usr/local/bin:$PATH

Solves the problem. Now the command bash finds the executable inside /usr/local/bin before it gets to /bin.