WOL with PCI card

I've got a Biostar TA790GXBE motherboard with onboard Realtek RTL8168D NIC. This has three boxes under Power Management:

Allow the computer to turn off this device to save power
Allow this device to wake the computer
Only allow a magic packet to wake the computer

I've added a Realtek 8139B NIC which only has the first option with the second greyed out and the third not present. However, under Advanced it has:
Shutdown Wake on Lan: Enabled
Wake on Lan Capabilities: Magic Packet & Pattern Match
Wakeup on ARP/Ping: Enabled
Wakeup on Link Change: Disabled
Wakeup using APM Mode: Disabled

I want to connect this NIC to the router for Internet access, with the onboard Gigabit NIC connected directly to another PC for RDP. If I run a WOL monitor, I can see it receives the Magic Packet on this interface, but it refuses to wake from shutdown, hibernate or standby.

In the BIOS it only has two relevant options:
Power-on by PCIE/onboard LAN
Wakeup by PCI

Is there any way to get WOL working with this card or will I have to give up on this idea?

Have you tried installing the newest driver?
Driver link: Realtek

Thanks, but that's one of the first things I tried.

OK, so you have the Realtek 8139B NIC in one pc and it is connected to the NIC in your other pc and the pc with the Realtek 8139B NIC will not wake up when a magic packet is sent, is that correct?
If so do you have a crossover cable connected between the 2 nics or are you using a straight through cable?

Not quite. The 8139B NIC is connected to my router (the onboard Realtek RTL8168D is currently not connected to anything) and the PC will not wake up when I send it a Magic Packet (although the WOL monitor shows that it's receiving it)and the "Allow this device to wake the computer" option is greyed out.

This is the reply I got from Realtek. I don't really understand how I check/enable ACPI for the PCI slot. Certainly the PC is listed as ACPI under Device Manager and I've enabled Plug & Play OS in the BIOS.

"Thank you for your E-mail. Do you insert 8139B card on PCI slot? If yes, please check ACPI mode
for PCI slot. If BIOS doesn’t support ACPI , item 2 & item 3 couldn’t enabled.

1.Allow the computer to turn off this device to save power
2.Allow this device to wake the computer
3.Only allow a magic packet to wake the computer"