An old trick to fix scratches in glasses was to rub in a little clear nail polish. I feel like if you can feel the scratches with a finger nail, no amount of toothpaste will get it out.

This is actually a pretty fantastic idea. The only problem I can see you running into is that those kits are usually designed to be used for chips, not wide scratches. Might be worth a shot to experiment though.

I would say start with some clear nail polish. Do you have a wife or daughter you can steal some from? In a worst case scenario that it looks terrible, you can always remove it with acetone. Then I'd probably consider moving onto the permanent solutions--like a windshield kit or something like that.