As others have said, exhaust rather than intake.

Use a 230MM exhaust fan to pull as much out as possible, and if you’re willing, add two small 50MM case fans to create intake at the bottom.

You could also stick a bowl of cold stones near a natural air intake to the cabinet during gaming sessions.

Additionally, if there’s a way to command the Xbox fans to run at a constant speed rather than ramping when needed, it will allow for a more continuous temp control.