I'm going to label a happy hour as invite only and secretly using my work email
Why would you label it that way?
I want to make my life more comfortable.
Understand that doing this may cause some people to feel excluded. If your intention is not to upset anyone, you should probably invite anyone who wants to come.
People need to get better about it, a happy hour is a social event. Sometimes it is just for a select group. I find that to be fine in my opinion.
Just try to put yourself into the shoes of the other workers. It would stink to not be invited to something like this if you wanted to go.