Will a backdraft damper in a air return duct cause issues with the gas furnace?

In our attic is a gas furnace for the 2nd floor. The attic is not insulated (except for the attic floor which is the 2nd floor ceiling). The air return for this furnance is in the 2nd floor ceiling on the hallway (looks like a rectangle, approx. 20ish inches in diameter). Cold air is leaking out of this air return vent. I assume it’s not attic air that leaks into the 2nd floor through this return duct, but the hot air rises into this duct, cools down and then sinks into the 2nd floor. I insulated the return duct in the attic with HVAC insulation which made things better. Now I am wondering if a backdraft damper would improve the situation. The flexible hose from the damper has 13 inch diameter and I bought a fitting backdraft damper. It opens easily and shouldn’t put too much pressure on the system. Could this damper cause any other issues than putting more load on the fan? The gas heater is 3 years old and a high efficency unit.

Edit – some more information:

The cold air from the return duct is more than 10 degrees cooler than the 2nd floor temperature. Right now it’s around 40 degrees in the attic, 70 degrees on the 2nd floor and the air from the vent is around 59 degrees.

You can’t feel much of an airflow from the duct. My guess is, that there’s no attic air entering the duct, but the hot air from the 2nd floor is rising into the return duct, cooling down and falling into the 2nd floor again.

My hope is, that the backdraft damper keeps the rising air in the duct itself, so it won’t travel through the round hose to the furnace. This hopefully keeps the air in the duct quite warm. The hose that goes from the return duct in the attic to the furnace is insulated and is around 12 inches in diameter and maybe 8 feet long.

Since the damper opens easily my hope is that the system won’t even notice the additional resistance.