Partial Controlled Residential Ventilation / Retrofit Climate in the Attic

We bought a small house from 2014 last year. A split air conditioner was installed in the living room on the ground floor; it works great and even last year, at nearly 40 degrees, had no problems reliably cooling the entire ground floor. We would now like to tackle the attic. We came up with the idea of installing a controlled residential ventilation system in the attic. The ceiling is wood/drywall. Core drilling between the rafters is possible. Additionally, maybe a small effect could be felt on the ground floor through the open staircase. Perhaps it would also be possible at the same time to cool the attic and distribute the cool air into the bedrooms? Would this work, or is the project doomed to fail? Here are two floor plans. We would completely exclude the basement. Perhaps a duct could also be realized via the existing but unused second flue in the chimney in the living/dining room?

Warm air rises. So you will hardly achieve any cooling effect through the controlled residential ventilation, probably rather the opposite. I would install another air conditioner in the attic and also a controlled residential ventilation for fresh air.

You might also have possibilities to get fresh air down not only through the stairwell.

Thank you for your response. I had not planned to use the controlled ventilation system to bring air up from below. It is already planned to install an additional air conditioner upstairs. However, we do not want an indoor unit in every room. That is why the idea is to distribute the cool air from the air conditioner upstairs. One possibility might also be to cool the attic and distribute the air with the controlled ventilation system?

With controlled residential ventilation, you cannot distribute anything.

If the controlled residential ventilation system sucks in the cool air at one point and this air is led outside via the heat exchanger, the sucked-in air should be cooled as well. Nothing else happens in winter with the heated air, right?

Unfortunately, this only works theoretically in the case of cooled air. The heated air in winter is constantly reheated in the corresponding room.

As I read your considerations, you want to distribute the cooler air into the rooms, and that simply does not work. As soon as the air is in the room, it will take on a homogeneous state with the air already present and, if necessary, will be reheated directly to room temperature via solar infiltration.

So, simply put: You cannot distribute anything with [Kontrollierte-Wohnraumlüftung].