Cooling with heat pump via underfloor heating?

When I asked what you would advise me, the first thing was tempering via underfloor heating and the up to 4 degree temperature difference.

I don't think your goal is to achieve "up to" (at least not mine) and also not only in the first few days (that's why it says up to). Once the sun has been shining on your house for a few days, these "up to 4" often become only "up to 2," and combined with the existing indoor humidity, it can really only be described as a placebo.

Insulation neither lets heat in nor out,

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

For us, the temperature control via underfloor heating costs around €4,000. The most expensive part is probably the switches. We have it included in the price. In addition, we will process it through BAFA. We should then get 35% back, I think.

Phew... €4,000 would be too much for underfloor heating cooling for me. With you, it would be correspondingly less through BAFA. But that is not yet certain for us.

When we built, I also considered the air conditioning more of a luxury. Now that the first summer in the house is underway, I am really glad to have it. Because it is very difficult to keep the heat out in such a well-insulated, new house. Already just the sunlight coming through the windows heats up the house. You either shade so thoroughly that you could live in a cave, or you let the light in. The amount of heating energy that 1 meter of window front with sunlight provides is not to be underestimated; I massively underestimated this during construction.
And once the warm air is 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 through ventilation.

Without going too much into a fundamental discussion and going down the eco-route:
But I do wonder if all this can really make sense. When you already appreciate the air conditioning running on a lousy summer like this year because otherwise, it would hardly be bearable as the window surfaces heat the house unbearably. I am not surprised that a Hofreiter no longer sees the single-family house as a model for the future.
If this catches on, future energy standards will focus much more on effectively preventing the house from heating up. Huge window areas facing south simply cannot be the goal anymore.

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?