Unless there is some piece you didn't mention in your original post (like the reason you need a serial port is for some old\legacy remote controller) what does the serial port have to do with sending video to a TV? You would do that with a video out, not a serial port. Like a VGA connection.
On a side note I'm impressed to hear of a Satellite 100 still being running. If I recall, that machine was new in the summer of 1996.