Consultation for photovoltaic system

... that only works if you really fully charge and discharge it 200 times a year. The former requires a photovoltaic system that is at least three times larger (in kWp) than the battery (in kWh). The latter actually only works with absurdly high night consumption in summer.

yes,
that is unfortunately correct.

The conclusion that one should just push electricity consumption at night to make it
computationally cheaper would not be correct.

That would be a fallacy because once the system is in place, every
(partial) charge cycle costs battery lifespan.
But it is hard to predict, as lifespan depends on many factors.
I would never assume a lifespan of 7,000 full cycles.
In my opinion, this is rather a theoretical idealized value, which
- much like the fuel consumption figures of car manufacturers - is practically never reached in reality.
The cells also age in standby, but especially at too high temperatures.
So preferably operate the battery in the basement, not in the attic (where the cable routing
might be easier).

Therefore, I view the claim of up to 15 years lifespan very skeptically.

Sizing the photovoltaic system for off-grid operation (full supply) is a very ambitious goal.
Generally, it does not pay off, also because of the absurdly high ecological upfront costs.

It is worth taking a close look at the warranty conditions. There are manufacturers with a 10-year full warranty that is cycle-independent. That can let you sleep a bit more peacefully. The subsidy in Bavaria, for example, requires a guaranteed performance of 80% after 10 years and corresponding warranties.

how many cycles are you currently doing with your battery per year?

I can’t say exactly, as I haven’t had the new system for a full year yet; I only had the first full month in the current combination in May.

From May 1st to August 31st, I come to 127.7 cycles.

The values below I copied from my portal. In our house, there are a few non-standard consumers, so the values should not be considered universally valid. We (still) do not have an electrically powered vehicle (which I also wouldn’t charge from the battery, but rather use surplus instead of feeding in).
Another factor could be that the system with 8kW charging/discharging power is less sluggish than many other systems and therefore, especially during variable weather with some sun, feeds a lot into the storage and can also supply or co-supply heavier consumers when discharging.
The losses of just under 20% in the system can be seen in the difference between charging and discharging. The loss values are not constant in percentage terms, which I cannot fully explain, except that month transitions with different states of charge could play a role.
We have 8kWh gross battery capacity – so it should be about 7.2 kWh net capacity. I cannot directly assign the cycle numbers of the batteries to a period in the portal. Therefore this auxiliary calculation:
May:
charged: 214
discharged: 177
214 / 7.2 = 29.7
June:
charged: 237
discharged: 205
237 / 7.2 = 32.9
July:
charged 238
discharged 198
238 / 7.2 = 33
August:
charged 231
discharged 193
231 / 7.2 = 32.1

But somehow that doesn't make any sense, does it? You would have completely drained the battery more than once a day. And that in the summer.

Or am I making a mistake in my thinking here?

Addendum:
In the above-mentioned 4 months, I withdrew 693kWh from the batteries.
At a purchase price of 27ct minus 11ct feed-in tariff, this corresponds to 693 * 0.16 approximately 110€. With a feed-in tariff of a new system with today's registration of 7.25ct, that would be 693 * 0.1975 approximately 137€