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?

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