Help with room ventilation

[Gocntry]

I have a 12" box fan in there that is always on just to circulate the air. I thought about cutting a hole in the door and putting a metal vent in it towards the top. Like the kind you would normally have in a closet that has a hot water heater or furnace in it. Would that let too much air out? If it were just a stationary vent that wouldn't open or close?

You could put something like this in the door, one in the top and one toward the bottom

https://www.lowes.com/pd/Accord-Vent...-in/1000033685

Then you could control how open or closed they are and with a box fan blowing out thru one it should pull air in thru the other.

If the outer room has a/c it might pull in cooler air while pushing out how air, and you can close them up in the winter if needed.