I have devised a solution for this problem with the assistance of the Esri ArcGIS Pro SDK team. I am utilizing the GeometryEngine.Instance.Move method on the polygon along with a delta X and delta Y coordinate. The previous mouse point X/Y is subtracted from the current mouse point X/Y. This creates the delta that is needed to move the centroid of the polygon to the correct location. The following is the necessary code:
MapPoint screenMapPt = MapView.Active.ScreenToMap(new Point(e.X, e.Y));
MapPoint prevPt = MapView.Active.ScreenToMap(new Point(_prevX, _prevY));
ArcGIS.Core.Geometry.Geometry buffer = GeometryEngine.Instance.Move(_graphics[i].Polygon, (screenMapPt.X - prevPt.X), (screenMapPt.Y - prevPt.Y));
_graphics[i].Polygon = (Polygon)buffer;
The code is still a little bit laggy, since it all happens on the mouse move event, but it is faster than the previous option of creating a GeodesicBuffer.
Thanks again to the Esri ArcGIS Pro SDK team.