The batting can make an amazing difference. Sometimes bright white is better, sometimes natural is better... all depends on the fabrics used and the look you want.
I'll do a lot of things, but I don't think I'm willing to deal with bleaching a batting... but I would definitely audition a few options before deciding on the final choice.
Edit: you might get some idea by putting down a white piece of fabric and then laying the top over that compared to a neutral batting.