Update SharePoint external data column (connected to SQL) via CSOM C#

Question

I need to add a document to a document library (on-premise SharePoint 2016) and update the metadata. I have pieced together the code to add a document, as well as successfully update multiple choice fields, but I need to update an external data field which SharePoint receives from a SQL database. Is this possible?

My update code looks like this:

    public class MetaData
    {
        public string Field { get; set; }

        public string Value { get; set; }
    }

    private static bool UpdateDocument(string fileName, List<MetaData> metadata)
    {
        bool _result = false;

        // Create context
        using (SP.ClientContext _context = new SP.ClientContext(_url))
        {
            // Get library
            SP.List _library = _context.Web.Lists.GetByTitle(_libraryName);

            // Get root folder of library
            SP.Folder _rootFolder = _library.RootFolder;

            // Load data
            _context.Load(_rootFolder);
            _context.ExecuteQuery();

            // Caculate file url
            string fileUrl = string.Format("{0}/{1}", _rootFolder.ServerRelativeUrl, fileName);

            SP.File _file = _context.Web.GetFileByServerRelativeUrl(fileUrl);
            _context.Load(_file);
            _context.ExecuteQuery();

            SP.ListItem _item = _file.ListItemAllFields;
            _context.Load(_item);
            _context.ExecuteQuery();

            foreach (MetaData _mdata in metadata)
            {
                SP.Field _field = _library.Fields.GetByTitle(_mdata.Field);
                _context.Load(_field);
                _context.ExecuteQuery();

                _item[_field.InternalName] = _mdata.Value;
                _item.Update();
                _context.ExecuteQuery();
            }

            _result = true;
        }

        return _result;
    }

FYI: The field I need to update is called 'Client'

I can set the field itself, but of course, the value is not really 'there', although it is displayed. I've read several posts on how to do it from a machine running SharePoint, but there has to be a way to do it from a client.