(1) Try updating the Drivers for the second Real Tek card to see if an advanced driver adds the Power Management tab.
Typically Wired NICs do not turn off to save power. None of mine at any time.
(2) Look in Windows Advanced Power Management settings for the network cards. Try setting Power Properties to Maximum Performance. See Screen Shot.
(a) as noted LAN cards do not go to sleep.
(b) Windows 10 and Vendors are changing power management settings. So look where I described and look also in Start, Settings, System, Power and Sleep. These settings also depend on the model of PC / Laptop.
Just to belt and suspender: If the LAN card manufacturer will allow the LAN card to go to sleep (I have never seen this), they must provide an option to control sleep. If the LAN card will not go to sleep (what I have / what I have seen) then a power option is not needed.