I can see why you are asking for suggestions. You have already looked at the most obvious items. I would like to propose a possible reason. Maybe one of the voltages in the server is slow to ramp up or has some strange glitch when the server wakes. This could be caused by a power supply component or by any hardware component that is installed in the server. I know this theory sounds "out there, " but many years ago I did hardware maintenance, and I can recall some weird situations where some component other than the power supply made the power supply look bad. And less than perfect power can cause all kinds of weird symptoms.
I like your idea of swapping the network adapter. If the Windows error is caused by some condition that is on the edge of working correctly, you might end up with both computers coming out of sleep normally after the swap. Of course, if the wake error follows the adapter, then you just need to replace the adapter.
If the problem stays with the server, then tracking down the source will be a lot more work. You could swap other components, but first I would double check that the BIOS settings are the same (and the same version of the BIOS is in both), double check driver versions, and other settings.
I am very curious to hear about what caused the problem once you track it down.
Mark H