How do I specify NetworkService or LocalService for a Windows Service on a Domain Controller?

Based on your final screenshot, you left out the space in the account name. Try Local Service instead.

If that doesn't work, you can do it on the command line as shown here, I would write this as:

sc config ServiceName obj= "NT AUTHORITY\Local Service" password= ""

Note that in the command line version either LocalService or Local Service is acceptable, but in the GUI only the latter works. I'm not sure why, but that's the way it is.

You need to add "Built-in security principal" to your Object Types when doing this on the Domain Controller. As it is, you're only searching for User and Service Account Object Types.