Microsoft.WindowsAzure. Storage.StorageException: The client could not finish the operation within specified timeout

Maybe you can try to disabling dashboard logging. For this you need to put following code:

var config = new JobHostConfiguration();
config.DashboardConnectionString = null;

WebJobs SDK stores log blobs to “azure-jobs-host-output” container for each execution of the method. In your scenario maybe the process uses lot of bandwidth and run into timeouts from the SDK trying to save log blobs.

Reference:
https://github.com/Azure/azure-webjobs-sdk/issues/1393