Mike,
The labelpoint method on the GeometryEngine class should do it.
The following code will find the labelpoint for the first selected feature.
protected override void OnClick()
{
//Run on MCT
ArcGIS.Desktop.Framework.Threading.Tasks.QueuedTask.Run(() =>
{
//get the labelpoint for the first selected feature
var selectedFeatures = ArcGIS.Desktop.Mapping.MapView.Active.Map.GetSelection();
//load into editors inspector to get the shape geometry
var insp = new ArcGIS.Desktop.Editing.Attributes.Inspector();
insp.Load(selectedFeatures.First().Key, selectedFeatures.First().Value.First());
var selGeom = insp.Shape;
var labelPoint = ArcGIS.Core.Geometry.GeometryEngine.LabelPoint(selGeom);
});
}