Although it seems to be a conversation with myself some follow up:
What I could not get out of my head was, that it only occurred on battery. Sooo i did some more digging. I tried the Crucial P3 from my wife and it worked, so I guess it is no hardware error on my mainboard.
Than I tried different PowerState limits but nothing worked. Firmeware of my Crucial P1 was also the latest. Tried forcing the same PCIe PowerManagement on AC and Battery pcie_aspm=force pcie_aspm.policy=default did not work either. So I gave up. Left the nvme_core.default_ps_max_latency_us=0 boot parameter and set up hibernation as a power saving solution. That worked.
Than I found this: NVME timeout woes - #9 by Chris_J and now it seems it comes down to a kernel regression since 6.7.10
I was new to Ubuntu, so I have never had an older Kernel to experience my nvme working properly but that is f***ing amazing.
EDIT: It only took longer but still happend after waiting long enough