Chatter - we need to completely remove the "All with access" feed option from the internal view in SFDC

As we work to implement an externally facing SFDC community, we need to be able to ensure zero opportunity for our internal users to inadvertently post something that will be viewed publicly when they thought it would be private.

The only way I know to do this would be to disable chatter in the external community, thereby eliminating a lot of the potential community value. We’d like to keep chatter on specific objects, but want to ensure no chance of crossing the streams. With thousands of employees, objects, and interactions, if it can happen, it will happen.

So specifically, I want there to be no ‘All with Access’ option appearing in the internal Chatter widget on an object for internal users or show "All with Access" only to certain profiles or when they select "All with Access" it can display the popup or any other options.

Is this possible?

If not what other options I have like Customization.