Yes Deployment manager should work in this situation although it is not something that we officially support.
I believe that you are trying to publish to the embedded nuget feed that is installed with Deployment Manager. To do this you will need to change a registry key value, using regedit.exe. You will need to change the value of the EmbeddedNuGetUri String value key in [HKEY_LOCAL_MACHINE\\SOFTWARE\\Red Gate\\Deployment Manager] to https://deploy.mycompany.com:443/nuget/
and then restart the Red Gate Deployment Manager service in Services.msc.
The Visual Studio plugin asks Deployment Manager for the list of feeds, which it obtains from the information stored on the Package feeds page. You may also need to change the value listed here to match the value that you put in the registry although the service restart should do this. The Package feeds page can be accessed via the Settings tab.
Please be aware that if you run the Server Tools application, this will reset the bindings of your deployment manager server and the embedded Nuget feed.
Hope that helps.