Here's my solution. If anyone has a better one, I'm all ears!

using System;
using Umbraco.Core;
using Umbraco.Courier.Core;
using Umbraco.Courier.Core.Diagnostics.Logging;
using Umbraco.Courier.Core.Enums;
using Umbraco.Courier.Core.ProviderModel;
using Umbraco.Courier.ItemProviders;

namespace MyNamespace.Umbraco.Events
{
    public class EventSubscriber : IApplicationEventHandler
    {
        private void UpdateDocumentCache(object sender, ItemEventArgs e)
        {
            //this is Courier Item, not a native umbraco document
            var d = e.Item as Document;

            //the current provider used 
            var provider = sender as ItemProvider;

            //queue the event "UpdateDocumentCache" in the DBtransactionComplete queue
            //this will then be executed after the DB transaction is done.
            if (d != null && d.Published && provider != null)
                provider.ExecutionContext.QueueEvent("UpdateDocumentCache", d.ItemId, null,
                                                     EventManagerSystemQueues.DBTransactionComplete);
        }

        #region IApplicationEventHandler

        public void OnApplicationInitialized(UmbracoApplicationBase umbracoApplication,
                                             ApplicationContext applicationContext)
        {
        }

        public void OnApplicationStarting(UmbracoApplicationBase umbracoApplication,
                                          ApplicationContext applicationContext)
        {
        }

        public void OnApplicationStarted(UmbracoApplicationBase umbracoApplication,
                                         ApplicationContext applicationContext)
        {
            DocumentItemProvider.Instance().PostProcessed +=
                new EventHandler(UpdateDocumentCache);
        }

        #endregion
    }


    public class CourierUpdateDocumentCache : ItemEventProvider
    {
        public override string Alias
        {
            get { return "UpdateDocumentCache"; }
        }

        public override void Execute(ItemIdentifier itemId, SerializableDictionary parameters)
        {
            Guid g;
            //if the itemId is a valid Guid we will fetch a umbraco document from it
            if (Guid.TryParse(itemId.Id, out g))
            {
                try
                {
                    //get the native umbraco document object
                    var d = new umbraco.cms.businesslogic.web.Document(g);
                    var cs = ApplicationContext.Current.Services.ContentService;
                    var c = cs.GetById(d.Id);
                    cs.Publish(c); // let the magic happen
                }
                catch (Exception ex)
                {
                    //logging of errors
                    RevisionLog.Instance.AddItemEntry(itemId, this.GetType(), "UpdateDocumentCache", ex.ToString(),
                                                      LogItemEntryType.Error);
                }
            }
        }
    }
}

Coincidentally, there is one line that is using an obsolete call (var d = new umbraco.cms.businesslogic.web.Document(g)), but I didn't spend a great deal of time to figure out how else to take the Courier guid and translate to a Content object since IDs are numeric.  When I look at the Document object it's executing a direct SQL scalar call, so no translation to the preferred object like many other obsolete references.