Doors – How to weather strip a door in an outside building

The fact that you have this problem even though you keep the office door open indicates that your problem is an insufficient supply of conditioned air to the room, rather than the lack of a return path for the air. This can be caused by the layout and sizing of your ductwork.

Before taking on this project, you might try closing off the vent(s) in another room to force more air into your office. If you have a room that stays too cool in summer that would be an ideal candidate. If that works and you want to make the change more permanent, you could install an adjustable damper in that room's duct and set it to a position that keeps that room comfortable but still allows a bit more flow to the office.

If that doesn't work out then I think your proposed solution is a good one, possibly the only one short of a major overhaul of your ductwork to get the airflow more balanced. See this question for some tips on installing such a duct.

When you have to fill the gap, use stainless steel or copper wool (not kitchen pad with soap though) and optionally foam the gap closed. The metal stops the mice from eating through the foam.