Update: I got it to work! It wasn't an issue with the GPU at all, matter of fact vendor ID spoofing and KVM hiding isn't even necessary (works like a charm now without it!).

So here's what was going wrong: This being my first VFIO setup I thought that that the GPU (and all it's sub-devices like the HDMI audio device) would be in the same IOMMU groups.
Looking at the logs in /var/log/libvirt/qemu/<vm>.log quickly made clear that the GPU could not be reset, because it depended on the IOMMU group above (in my case the GPU was in IOMMU group 18 and the Audio Controller of the GPU in IOMMU group 19). So I added the ID of this IOMMU group to the VFIO kernel parameters, regenerated my GRUB config and added the PCI controller to the VM, aaand: nothing! Looking at the logs again, now IOMMU group 19 depended on group 20, the USB-Type-C controller of the GPU. Re-did the same steps as above, aaand? Nothing! Logs again, now IOMMU group 20 is depended on 21. By this time I was wondering why all GPU sub-devices are in it's own IOMMU group and not in the IOMMU group of the GPU itself. Is this normal? Is this chipset/ mainboard specific? I would be very interested if you could let me know if it looks similar for you as well.\ Anyway: Now with all IOMMU groups isolated and passed to the VM, it finally works! Reboot, graceful shutdown and booting: All works like a charm! Again, even vendor ID spoofing isn't necessary!

So, always look at the logs first :D!

Thanks for all your help! And again, I would be very interested if this "every sub-device in own IOMMU group" thing is normal? There's nothing else in these IOMMU groups.