My dependent was denied because she grossed more than $4,400. That’s barely anything to make in a year, does it mean $4,400 per month?

No, it’s per year. 

It may not be much comfort, but even if you could claim your GF, all you get is a $500 credit that could reduce your tax liability.  No added refund at all.

The rules for claiming an unrelated person as a dependent are much more strict than the rules for claiming a related person.  You would have to persuade Congress to change the rules.