Previously, i have tried to use private Nuget feed hosted on Azure DevOps but face the issue to pass credentials to access it from Docker and perform the restore without nuget.exe.
It turns out the point has since been solved by Microsoft.
The solution is to use a this tool made by Microsoft and pass a Personal Access Token (PAT) in the Docker. I use a multi stage build so the credentials will be ephemeral.
The Github provides a sample on how to use it.
Hope it helps !