I am also using AGSLocalTiledLayer. The max scale supported by my TPK file is 5000, but I am looking to achieve the scale 1250. I was able to zoom in to the scale 1250 by setting the maxscale = 1250 to both my localTiledLayer and mapView. Doing this, I am able to zoom to scale 1250 (The last tiled image is zoomed/pixelated). I think we should be happy that the last tile is pixelated instead disappear altogether. I am not sure if this is what you were looking for.
My other requirement is at the scale of 1250, when the user pans the map, the tile available at 5000 (last tile) should be loaded with pixelated view. Right now, when I pan outside my visible extent at 1250, there is no background/base map.
Do you have any thoughts or approach that I can try?