I could solve the issues on Fedora Silverblue 44 with Linux kernel 7.0.6.
The root cause is PCIe ASPM (Active State Power Management) putting the card (including the PCI-to-PCI Bridge) into a state it can’t recover from. The fix is to disable ASPM system-wide at boot by adding the kernel parameter pcie_aspm=off.
On the other side I still had issues when GNOME triggers the suspend after 15 minutes of idle. The fix is to simply disable the suspend on idle.
If there’s any better solution, please let me know.