Hi Leland.

So there is a Service call you can use that will clear the RecycleBin in the ApplicationContext.Current.Services.ContentService called EmptyRecycleBin().

What I would recommend is to create a UmbracoApiController method that clears the recylce bin. Here is a little POC on how your ApiController could look. The url for this Api method will be http://yourdomain.com/Umbraco/Api/RecycleBinCleanup/ClearRecycleBin.

using System;
using System.Web.Http;
using Umbraco.Core;
using Umbraco.Core.Logging;
using Umbraco.Web.WebApi;

namespace YourApplication.App_Code
{
    public class RecycleBinCleanupController : UmbracoApiController
    {
        [HttpGet]
        public string ClearRecycleBin()
        {
            try
            {
                ApplicationContext.Current.Services.ContentService.EmptyRecycleBin();
                LogHelper.Info<string>("RecycleBin Cleared at " + DateTime.Now);
                return "OK";
            }
            catch (Exception exception)
            {
                LogHelper.Error<string>("Could not emptyRecycleBin", exception);
                return "Error";
            }
        }
    }
}

And then create a Windows Scheduled task on your server that runs on an exact time for example every sunday. https://technet.microsoft.com/en-us/library/cc748993(v=ws.11).aspx#BKMK_winui

Or you could use Umbraco´s built in Scheduled Task, but those are based on intervals not exact time, but maybe that could work for you if you tweak the api method so that it checks the current dayOfWeek, or set it to a higher interval. Scheduled tasks are configured in the /config/umbracoSettings.config file in the scheduledTasks section. http://stackoverflow.com/questions/15415175/scheduling-to-run-a-macro-on-every-hour

<scheduledTasks>
    <!-- add tasks that should be called with an interval (seconds) -->
    <task log="true" alias="clearRecycleBin" interval="60" url="http://yourdomain.com/Umbraco/Api/RecycleBinCleanup/ClearRecycleBin"/>
  </scheduledTasks>

NOTE: The above task would actually clear your recycle bin every 60 seconds, and you probobly want a higher interval set, this was only set for testing.

I hope this was helpful. Let me know how it goes.

Take care!