As far as I know the "Register Existing Data" option is only available in the ArcMap UI on the Distributed Data wizard.
Right. That's the only place I've ever seen the option. We're just looking for way to perform this without ArcMap. The environment is fairly locked down, so we don't have connectivity to it from our normal desktops with ArcMap. And adding an ArcMap license to that system is superfluous as it would only get used if we had to rebuild the system. (Hence the desire to do it through ArcPy)
I know that they have looked into incorporating this in a future release but no official word on that release at this time. So the ArcMap option is the only way right now to do it.
The AddDataStoreItem arcpy functions exist for both Pro and ArcMap and can be used to automate/script the creation of file based data stores and database data stores: