Hi Brian,
I'm having the same trouble as you. It's a shame as I think HDMI is the only way of getting digital inputs to sony bravias (very dumb if you ask me given these are 'colour like no other' digital televisions!
The only thing I can offer you is that if you turn off 'overscan' in your Mac Mini's displays control panel, it'll shrink the screen so all of it's visible, but it will mean you've got black borders round the edge. I wound up making a few applescripts so you can switch quickly between overscan and no overscan if, say, you want to watch a DVD then want to browse the web.
I've gone back to a VGA cable at the moment, which seems to be the only way you can get a Sony Bravia to give you pixel-to-pixel matched picture. Sadly the contrast is nothing like as good as on the DVI connection. Wish I'd noticed this before buying the telly...