The CallMethod of the PrivateAccessor is using reflection to invoke the required method. When an exception is thrown inside that method it will be wrapped by the reflection in a TargetInvocationException which is received in the test method.
Instead of the TargetInvocationException, throw the original exception.
Docker is a container acting like an isolated environment. Research how the JustMock profiler can be registered into such container. Hosted VSTS should work on the same principle. Research how the registry could be accessed through VSTS extension or other tools.