Eventually I solved this problem using an application called "File Browser." Instead of sending files to the Video section of iTunes, use the App File Sync feature of iTunes to send the files to File Browser. Note that other file utility apps should work as well too, not just the one I use (called File Browser).
After moving the files to the file app, you can browse the files from the app. File Browser will start video files using the iPad's native Video viewing program. So it accomplishes the goal stated in my quesion.
Since I started doing this I now use a Windows program called Disk Aid which emulates iTunes to the iPad and allows you to have total control over the iPad's file structure for copying files to and from the device. I still use File Browser on the iPad to start the video files from the iPad. I use Disk Aid to move files to the iPad instead of iTunes, and I start video files on the iPad using File Browser, which then allows me to run the videos in the native iPad video program.