Cooling with heat pump via underfloor heating?

In the summer months, our ventilation system runs only at night. If I run it during the day, no matter how much I shade, I’m just pumping warm outside air into the house. Then there’s the heat recovery of modern systems – you can partially switch it off via bypass/summer cassette, but it doesn’t help when it’s 30 degrees outside.
The underfloor heating doesn’t run at all because the heating system is in summer mode. The air conditioning runs daily, on and off, and dehumidifies/cools. I would never want to live without air conditioning again; it’s quietly humming quality of life on the wall.

In the summer months, our ventilation system only runs at night. If I let it run during the day, no matter how much I shade, I end up pumping warm outside air into the house.

Can't the ventilation system cool the air a bit? As far as I have read, it is possible. Also, the underfloor heating would run, as it is supposed to cool. It is clear that tempering via underfloor heating is not nearly as effective as an air conditioner. But it is enough if you go from outside into a room that is 4 degrees cooler. Then you have already gained a lot for little money.

Although it has nothing to do with the actual topic:

The heat input through the central controlled residential ventilation with activated heat recovery is minimal!

With 23 inside and 30 outside, the delta T is 7 degrees.
At 100m³/h, this results in a maximum of 250W if the bypass is active.
With heat recovery at 90% efficiency, only 25W remain.
That is really negligible.

So do not switch off the central controlled residential ventilation even in summer and only activate the bypass at night!

We did not buy the cooling function of the system because I had read in online research beforehand that it can only lower the temperature by 2-3 degrees (if it works well), and also cold air does not rise -> cold feet problem.
There are apparently systems where ventilation is combined with air conditioning. We have a normal, central ventilation system. It does not have a cooling function. The cooling function of the heat pump (if we had bought it) would only cool via underfloor heating.

When we built the house, I considered the air conditioning more of a luxury. Now, with the first summer in the house underway, I'm really glad to have it. Because it is very difficult in such a well-insulated, new house to keep the heat outside at all. Even just the sunlight through the windows heats the house up. Either you shade so consistently that you could live in a cave, or you let the light in. The amount of heating energy that 1 meter of window frontage with sunlight provides is not to be underestimated, I massively underestimated this during construction.
And once you have the warm air inside the house in summer, it is difficult to get it out again without air conditioning. The only thing I can think of is not ventilating during the day, and then trying to get the warm air out at night by ventilating. But in a summer like we had in 2019 for example, it was often still around 30°C at night. It is very uncomfortable to sleep at those temperatures.

I had managed in May to have 28°C inside the house even though the outside temperature was only 18°C. I hadn't shaded the windows and the ventilation system was running without the summer cassette, meaning with 98% heat recovery. Over several sunny days it got warmer and warmer inside the house. We only got rid of the warm air again when we manually set the ventilation system for summer (some exhaust air during the day, ventilate at a higher level at night) and already in May cooled down again with the air conditioning units.

Also with an enthalpy heat exchanger? Because we will have one of those too.

As far as I understood it as a layman, it can regulate the humidity.

Can the ventilation system not cool the air a bit? As far as I’ve read, it is possible.

Besides, the underfloor heating would be running, since it is supposed to cool. It is clear that tempering via underfloor heating is not nearly as effective as an air conditioning system. But it is enough if you go from outside into a room that is 4 degrees cooler. Then you already gain a lot for little money.