Maybe try iChat?
Go to the “VIDEO” menu and pick “Video Preview.” It should think your converter is an iSight camera and show it to you. You can’t make it full-screen in the sense that everything else is blacked out, but you CAN grab the corner of the window and drag it as large as the screen. You’ll see the desktop on the edges but it will be as large as you can get.
The only issue with this is that everything will be reversed like a mirror. You can solve that for $8, however:
https://www.ecamm.com/mac/iglasses/
That’s a cool little program that lets you change the video from an incoming firewire camera. Things like color corrections and brightness. And yes, you can have it flip the image, thus putting everything back correctly.