I used to ride a pony that had the same issue when ever we rode over grass. If she pulls down, give her a big 'argh' and held the reins firmly so she yanks herself in the mouth. When she brings her head back up and stops going for the grass, give her some praise and move on. Depending on how determined your horse is, she may take a while to teach. The first couple of times she might take you by surprise, but after a while you can feel when she is about to take a dive. Therefore you can tell her no before it happens, or get a firm grip on the reins.
I hope this works, and you should soon be able to go back to a lighter bit.