How to split a Polyline Geometry at a certain point in ArcObjects?

I would probably use IPolyCurve.SplitAtPoint and get the paths (parts) created after the split and do your processing on those. As long as the createPart argument is true, SplitAtPoint doesn't just add a new vertex, it inserts a new part between the split point and the end of the path.

_{(source: arcgis.com)}

Whether this is the best approach somewhat depends on what processing you need to do however.