There are several approaches to adding a ListViewWebPart to a page floating around the Interwebs, and I do believe over the course of the past week I have tried every one of them!  What I intend to do here is to cut to the chase and present the method that worked reliably for my purposes.

Scenario

As the title suggests, the scenario is that we are adding a ListViewWebPart to a page, utilizing one of the current list views as a "template", but making some changes for the purposes of this web part.

Solution

I recommend hitting the "view source" button on the syntax highlighter, I really need to switch to a wider blog template :-(. In any event, here is the solution that I employed, I added numerous comments to describe why things were being done. Hopefully it saves someone some time.

SPLimitedWebPartManager limitedWebPartManager = null;
ListViewWebPart wp = null;
try
{
    // assume that you have a limitedWebPartManager variable containing an
    // instance of a SPLimitedWebPartManager class.
    // also assume that you have a list variable containing a reference to the
    // SPList that you are pointing this ListViewWebPart at.
    // create an instance of the ListViewWebPart
    wp = new ListViewWebPart();
    // convert the list GUID to a string, must include braces (ToString("B")) and be in uppers (ToUpper())
    wp.ListName = list.ID.ToString("B").ToUpper();
    // optionally set the title of the web part
    wp.Title = "I Love List Views";
    // add the web part to the limited web part manager.  when you do this, a "hidden" view will be created
    // on the list referenced by the list variable.  we will later get a reference to this hidden view and bend it
    // to our will.
    limitedWebPartManager.AddWebPart(wp, "whatever web part zone you want to add it to", 1); // the 1 is the index within the zone
    // you need to update the list because a view was just added to it
    list.Update();
    // now that the web part has been added we need to get a fresh reference to it from the limitedWebPartManager.
    // there are plenty of great ways to do this, in this example i will be using the low-tech approach of enumerating the
    // WebParts collection and checking for the type and title that matches our part.  i am quite positive there are more
    // efficient ways of doing this, but this code gets called exactly once every couple months, so efficiency isn't worth
    // the extra development cycles
    foreach (System.Web.UI.WebControls.WebParts.WebPart webPart in limitedWebPartManager.WebParts)
    {
        // check to see if the type of the web part is ListViewWebPart, and if so, check if the title matches our guy.
        if (webPart is ListViewWebPart && webPart.Title.Equals("I Love List Views"))
        {
            // cast the web part to a ListViewWebPat so we can start to tweak its properties
            var listViewWebPart = webPart as ListViewWebPart;
            // here is some of the magic.  the web part now has its ViewGuid property populated which
            // contains the Guid of the hidden view.  we get a reference to it here.
            var view = list.Views[new Guid(listViewWebPart.ViewGuid)];
            // here we grab a reference to a template view.  this may be the default view of the list,
            // or maybe you have another view that you created that you want to use.  you don't have to
            // do this, but in this scenario i wanted to copy the filter and sort settings from an existing view.
            var templateView = list.Views["Name Of Template View"];
            // now we drop all of the view fields in the hidden view
            view.ViewFields.DeleteAll();
            // now we are going to go through a string array and add each of the strings to the view fields collection.  again,
            // you don't have to do this, but if you want to change what columns are displayed in your ListViewWebPart, you will
            // need to muck with the ViewFields of the view.
            Array.ForEach(new string[] { "First Name", "Last Name", "Address" }, f => view.ViewFields.Add(f));
            // here we copy the Query straight from the templateView.  if you didn't have a templateView you could always
            // just create an SPQuery instance and assign it to the view.Query property.  no points will be taken off for doing that.
            view.Query = templateView.Query;
            // update our hidden view, and bask in the awesomeness of our customized ListViewWebPart
            view.Update();
        }
    }
}
finally
{
    // do some fun cleanup of disposable items.  if you are wondering about the .Web.Dispose() bit look for my
    // blog article on the memory leaks in the SPLimitedWebPartManager.
    if (limitedWebPartManager != null)
    {
        if (limitedWebPartManager.Web != null)
        {
            limitedWebPartManager.Web.Dispose();
        }
        limitedWebPartManager.Dispose();
    }
    if (wp != null)
    {
        wp.Dispose();
    }
    // i probably missed some .Dispose()s since I cobbled this code together in Wordpress and not Visual Studio
}