Do people with admin permission still see the print code?   Might not be a permission issue but a report set up issue.  

You would only see a list of print codes if there is more than one report/print code available for that user/function.

I setup a test user to match my end user in everyway. When I switch the functional profile back to one with all authorizations it works just fine. Selection Print codes shows up immediately.

There are 4 codes available, so it should show up.

There may be an access code that is restricting that user

Could you point me in the direction of learning more about access codes? From what I can tell nothing is changing with those in our configuration. I'm only changing the functional profile.

Check the report if it has an access code assigned to it. If yes, then make sure that in the functional profile that the profile has access to that code.  

That was helpful. Found the report in the report dictionary and found that it was tied to a function, not an authorization. That function was missing in the authorizations.