Frequent HttpRequestException when talking to Cosmos DB from Web App - web-services

Question

We have a web service (Azure App Service) deployed to Azure that talks to our Azure Cosmos DB via the standard C# SDK for Cosmos DB/Document DB.
Both - app service and Cosmos DB account/collections - are in the same resource group and in the same location in Azure.
For certain bulk operations where the web service performs a burst of requests to Cosmos DB we are frequently getting errors in the web service when talking to the database:
System.AggregateException: One or more errors occurred. ---> System.Net.Http.HttpRequestException: An error occurred while sending the request. ---> System.Net.WebException: Unable to connect to the remote server ---> System.Net.Sockets.SocketException: An attempt was made to access a socket in a way forbidden by its access permissions
at System.Net.Sockets.Socket.DoBind(EndPoint endPointSnapshot, SocketAddress socketAddress)
at System.Net.Sockets.Socket.InternalBind(EndPoint localEP)
at System.Net.Sockets.Socket.BeginConnectEx(EndPoint remoteEP, Boolean flowContext, AsyncCallback callback, Object state)
at System.Net.Sockets.Socket.UnsafeBeginConnect(EndPoint remoteEP, AsyncCallback callback, Object state)
at System.Net.ServicePoint.ConnectSocketInternal(Boolean connectFailure, Socket s4, Socket s6, Socket& socket, IPAddress& address, ConnectSocketState state, IAsyncResult asyncResult, Exception& exception)
--- End of inner exception stack trace ---
at System.Net.HttpWebRequest.EndGetResponse(IAsyncResult asyncResult)
at System.Net.Http.HttpClientHandler.GetResponseCallback(IAsyncResult ar)
--- End of inner exception stack trace ---
[...]
at Microsoft.Azure.Documents.Query.ProxyDocumentQueryExecutionContext.<ExecuteNextAsync>d__0.MoveNext()<---
Each of our ApiController instances statically allocates a single repository class, which in turn fetches a IReliableReadWriteDocumentClient instance in its constructor and holds it for its entire lifetime via
IDocumentDbInitializer dbinit = new DocumentDbInitializer();
Client = dbinit.GetClient(endpointUrl, myAuthKey, connectionPolicy);
So in my understanding we should use only 2 Document DB clients for our 2 repositories in the entire web service.
Things we've tried so far:
throttle the requests on the client during the bulk operation to less than 3/s
reduce the clients ConnectionPolicy.MaxConnectionLimit from default (50) to 20
increase the apps ServicePointManager.DefaultConnectionLimit
None of these measures significantly reduced the number of exceptions we're experiencing.
Any suggestions how to avoid this error in the first place?
Additional Cosmos DB SDK functionality to tune/configure/adapt for our use case..?