Cooling with heat pump via underfloor heating?

That is logically not true. Insulation delays the temperature equalization between inside and outside!

Yes, I already understood that ;-) I just wanted to point out that it is often mistakenly assumed that insulation only keeps heat inside. But it also keeps it outside, because insulation generally has no direction.

Huge south-facing windows simply cannot be the goal anymore

Due to the sun's path over the day and year, it is often the east and west windows that cause much more of a problem in summer if they are not shaded.
South-facing windows can often be shaded quite effectively with a bit of roof overhang.

But yes, shading always has the highest priority and should be planned properly.

The 2-3 degrees from the floor cooling can at most be seen as a supplement.
In my opinion, however, it is completely sufficient to have 23 instead of 26 degrees in living and sleeping areas in summer 23.

Unfortunately, you are mixing quite a few things.
I have calculated the heat gains from controlled residential ventilation for you.

Here’s an estimate for windows:
The heat gain from unshaded windows can quickly be around 200W/m² of window area.
With a nice lift-and-slide door measuring 5x2m, that alone means 2000W from just this one window, which is 100 times as much as from controlled residential ventilation with heat recovery. ;-)

In principle, this exactly matches your feeling.
Insulation neither lets heat in nor out, controlled residential ventilation also neither lets heat in nor out.
Windows, however, do let heat in well, if they are not shaded. So here by radiation and not by convection.

But from your statements, I assume you don’t have external blinds (Raffstores)?

We have roller shutters; we just hadn’t expected such a high heat buildup (as you just calculated) from the pure solar radiation through the windows in the first spring in the new house. We naively assumed that since the windows are heat-insulated, the house would neither let heat in nor out. We eventually understood how important shading is, but those first nice days in May caused a lot of heat inside the house.
As you already said, the house neither lets heat in nor out. So we had to actively cool and since then have paid better attention to shading. However, we still notice that the house continues to warm up in summer; we cool regularly. But we also don’t sit in a bunker; closing all the roller shutters makes the house very dark and dull. So we take a reasonable middle ground between darkening and cooling.

If I were to build again and had the money, I would probably invest in smart home and automatically controlled shading. This was not at all clear to me when building the first time, how important it is.

Would you keep the controlled residential ventilation running normally in summer? We don’t have a summer bypass, so it runs with heat recovery. If it’s 32°C outside, 24°C inside (due to the air conditioning), and I then run the controlled residential ventilation, it will get warmer inside again, right? The air cooled by the AC is transported outside, and the 32°C from outside comes inside? Then I have to expend energy again to cool it down. Or am I thinking about this wrong?

Without getting too much into a fundamental discussion and going down the eco-route: But I do wonder whether all of this can really make sense. When you’re already happy that the air conditioning is working during a crappy summer like this year, because otherwise you could hardly stand it, as the window areas heat the house unbearably. I’m not surprised if a Hofreiter no longer sees the single-family house as a model for the future. If this becomes the norm, future energy standards will focus much more on effectively preventing the house from heating up. Huge window areas facing south simply can no longer be the goal.

We don’t have huge window areas facing south. Just a normal veranda door and two moderately sized windows, one in the living room area and one in the kitchen. By the way, we haven’t drawn any energy from the grid for the air conditioning yet; we usually operate it during the day when the sun is shining and our photovoltaic system generates the electricity for it.

We don't have huge window areas facing south. Just a normal porch door and 2 moderately sized windows, one in the living room area and one in the kitchen.
By the way, we haven't drawn any energy from the grid for the air conditioning yet; we usually operate it during the day when the sun shines and our photovoltaic system generates the electricity for it.

And at night, when the photovoltaic system is not producing energy and there are still tropical nights, do you keep the air conditioning off?

We have roller shutters, we just didn’t expect such a high heat buildup (as you just calculated) from the pure sunlight through the windows during the first spring in the new house.

Yes, since retrofitting venetian blinds is usually very expensive and often not attractive, active cooling is probably the only option. Thanks to photovoltaics, though, it’s not quite so bad. ;)

Would you keep the controlled residential ventilation running normally in summer? We don’t have a summer cassette, so it runs with heat recovery. If it’s 32°C outside, 24°C inside (because of the air conditioning), and I then run the controlled residential ventilation, the house will get warmer again, right? The cooled air from the AC is transported outside, and the 32° outside air comes in? Then I have to use energy again to cool it down. Or am I thinking wrong?

Yes, keep it running, because that pretty much matches my figures. You need air anyway, and with heat recovery it’s only 25W of additional heat. With manual ventilation it would be much more! And your AC probably cools with 3000W or more. The one percent from the controlled residential ventilation won’t be noticeable in your power consumption, but you will definitely have better air.