Wether or not this window is shown is determined by a registry value.
(The version may be different on your system)
You could try to set this value to 0 in your extensions
Startup() method. If that's early enough I don't know. If not, maybe you could find another way to change the value.
Or perhaps you could instruct your users to start (double-click) the mxd instead of ArcMap. This will also skip the Getting Started dialog.