It's been my experience that electronics manufacturers design their products so no damage will be done before the fuse blows, but there's always a possibility. For $37, why not just send it in for repair? At that cheap, no sense in trying to make a DIY project out of it. Maybe they'll even warranty the repair, and test the inverter to make sure it functions correctly.
You could probably use 40 amp fuses with no issues, yes.