Heat pump is not compatible with a water-bearing fireplace

Basically, I believe the connection was better before than it is now.

No, "new" is better. Because this:

The heat pump should claim a heating circuit for itself. The photovoltaic system can support the heat pump well. Your fireplace can be useful for hot water generation. Running all three together through one buffer tank is generally difficult.

You can separate the heating circuit of the heat pump from the buffer tank as well. Then it just runs normally at first.

is nonsense.

The bivalent parallel heat generation with different flow temperatures and volume flows of the heat generators should ALWAYS be done together via a buffer tank, which simultaneously acts as a hydraulic separator. In manual or automated non-simultaneous operation, deviations are possible, but definitely not in this case. In doing so, all sources feed into the buffer and all sinks are supplied from the buffer.

The connections on the buffer should always be distributed from top to bottom over the entire buffer height in descending temperature order.
At the top the hottest, i.e., the supply line from the fireplace. This should always be around 70°C during operation. As can be seen in the drawing, the stove has a return temperature increase. The small circuit controlled by the 3-way valve there ensures that the water flowing through the fireplace is circulated in a loop until it reaches the minimum temperature and is only then "decoupled." This prevents the water in the stove from being too cold, so that no condensate forms on the heat exchanger at the stove and thus no rust occurs.
The line to the stove can also be the "old" one, which would better utilize the buffer volume.

The remaining connections also look good in terms of distribution by descending temperature.

Yes, we already had that last year. Then it bangs like crazy and it pumps cold water.

But when we turned it on last week, I throttled at 85 degrees and increased the pump flow rate so it wouldn’t happen anymore.

Automated stoves, such as pellet or gas boilers, have a switch that can be toggled and that shuts off when the target temperature is reached. The fireplace, however, contains fuel and the exhaust gases run through the heat exchanger. The fireplace cannot be switched off directly but is very sluggish.
At too high temperatures, the cold water injection starts, while simultaneously or immediately afterwards the overpressure valve triggers (that’s the "banging"), and the excess water goes into the drain. In this case, energy is lost into the drain, but that is intentional. This can be avoided if you only turn on the fireplace when heat output is guaranteed.

Your example shows very well why in a modern house with a heat pump, installing a water jacket in the chimney should be avoided as much as possible.

As a technically untrained layperson who has never worked with this, I would be cautious about it.

The question of course is how to get out of this situation easiest. To me, dismantling the water jacket sounds like the best solution and then properly adjusting the heating.

Dismantling is not necessary, just a careful approach as described above: only turn on the stove when there is demand.

Is it a room air-independent fireplace with air supply?

Much more important for the solution: what color is it? Red fireplaces tend to overheat more than green ones. Green color calms, especially as a spinach smoothie.

The problem is the buffer tank and not your . Mistakes were made here.

I am so against it when Mr. Drosten demands that only experts accredited beforehand by him/WHO/federal authority may comment on respective topics and everyone else gets silenced. Against this background, I welcome your interjection but would simply say: no.

The pipe to the oven can also gladly be the "old" one, then the buffer volume would be better utilized.

Thank you very much for your answers. We also asked our heating engineers about the "old pipe" to it, because we also consider that the most sensible, but according to him it is not possible, since the old connection would be its own piping circuit in the buffer…

These are already good news! Now we might have to wait a bit to see how the temperatures behave inside the house. Will it get uncomfortably warm or cold in certain rooms or even in the whole house? If so, adjustments need to be made specifically at the heating circuit distributors or the heating curve.

So unfortunately it is too cold in the bathroom now (21.5) and also in the living room without the stove… In the bathroom we would like 23-24 degrees. What should we do now? Raise the heating curve and target temperature?

So unfortunately it is now too cold in the bathroom (21.5) and also in the living room without the stove...
In the bathroom, we would like 23-24 degrees.

First of all, fundamentally: In (modern) houses, a large temperature difference between rooms is not (sensibly) possible.
This is simply because the insulation is only on the outside, but not between the rooms.
So 21 everywhere and 24 in the bathroom will hardly be (sensibly) possible.
However, 22 everywhere and 23 in the bathroom is perfectly doable! Maybe even a bit more difference.

What should we do now?
Increase the heating curve and set temperature?

I would first start by raising the set temperature and only adjust the heating curve if it starts getting cold inside during very cold outside temperatures.
It is important to first keep all the controllers in the rooms fully open.

First of all, fundamentally: In (modern) houses, a large temperature difference between rooms (meaningfully) is not possible.
This is simply because the insulation is only on the outside, but not between the rooms.
So everywhere at 21°C and in the bathroom at 24°C will hardly be (meaningfully) possible.
Everywhere at 22°C and in the bathroom at 23°C, however, is perfectly doable! Maybe even a little more difference.

I would first start by raising the set temperature and only then adjust the heating curve when it starts to get cold inside at very cold outside temperatures.
It is important to first leave all the controllers in the rooms fully open.

Okay.
We now have the rooms where we don’t need heating (storage, pantry, bedroom) set to 2, in the living room and bathroom to 6, and the rest at 4…

2, 4, and 6 are not standardized values. You have to see what exactly is behind them. Our thermostats (Busch-Jäger) operate in 5-degree increments, i.e., 2 equals 10 degrees, 4 equals 20 degrees, and 6 equals 30 degrees. One wonders what the developers were smoking when they came up with the idea...