If all else fails, you could install a PCIe NIC.
Does the BIOS provide Ethernet port (or LAN cable) diagnostics? If so, do they fail when the motherboard is cold and do they succeed when it's warm?
Are you 100% sure the motherboard doesn't short to the case or a mounting post when cold? It's a bit of work, but you could test it outside of the case to determine if the issue still occurs.