- create your map (with or without an extent, if you specify an extent, it needs to be in web mercator if you're using tiled services in web mercator)
- add a tiled web meractor service using ArcGISTiledMapServiceLayer
- add your dynamic map service using ArcGISDynamicMapServiceLayer
If your map service that has a spatial reference of 2278 is tiled, you cannot overlay it on web mercator tiled services because tiled services cannot be reprojected on the fly.
One general takeaway from this is that your map's spatial reference has to match that of your tiled services. The map's spatial reference is defined by the spatial reference of the extent you pass to the map constructor, or the first layer that is added to the map.