It looks like you're just using Basic Authentication and sending the username/password as part of the URI. Have you tried
using Elasticsearch.Net;
using Nest;
var nodes = new [] {
new Uri("http://1.2.3.4:9200/")
};
var connectionPool = new StaticConnectionPool(nodes);
var connectionSettings = new ConnectionSettings(connectionPool)
.BasicAuthentication("username", "password");
var elasticClient = new ElasticClient(connectionSettings);
Since you only have a single node, you can also use SingleNodeConnectionPool
var connectionPool = new SingleNodeConnectionPool(new Uri("http://1.2.3.4:9200/"));