ansible tower_project module, always getting "ansible-tower-cli required for this module"
Found the origin of the issue, and unless someone else had run into it they wouldn't have known where to look --
check details of tower-cli --version and ansible --version, as well as which tower-cli which ansible;  in my case were installed in different python environments:

/usr/lib/python2.6/site-packages/ansible
/usr/lib/python3.4/site-packages/tower_cli

If you cannot see the python version using the above for tower-cli, you can run:
$ head -n1 `which tower-cli`
#!/usr/local/opt/python/bin/python3.7
$ 

The ansible modules couldn't see the tower_cli python modules as they were in a different python environment.
After-the-fact problem origin:  I had installed ansible via yum (package manager), and ansible-tower-cli via pip (python installer).  Being consistent in how you install both to make sure they go to the same environment resolves the issue.
Centos/RHEL: yum install ansible ansible-tower-cli
-OR-
Python:  pip install ansible
Python:  pip install ansible-tower-cli