Hello Brian,
Just to clarify, are you having issues getting both basemaps to show up in your application? If so, the issue might lie in how you cached your own basemap. In order for these two cached services to run side by side, they need to use the same tiling scheme.
To check this, you can open up both REST endpoints (ESRI's cached basemap service and your own cached basemap), and compare the details found under "Tile Info:". These need to match.
When publishing my own cached basemaps to work with ESRI cached basemaps, I always select "ArcGIS Online/Bing Maps/Google Maps" for the tiling scheme. This seems to match up with what ESRI uses, and makes the basemaps usable together.
I've attached a jpg showing where to set this in the service editor. Hope this helps[ATTACH=CONFIG]29487[/ATTACH]