I would borrow bits and try them before buying. Go with the gentlest bit you can, a fat french link. If that doesn't work, go a little thinner. Don't automatically go to a harsher bit, if things aren't working for you. She may be already fighting a bit that is hurting her. Dutch, or three ring, gags are tough, if on the bottom ring and thin is even tougher. Do you ever ride with the reins on the main top ring?
She said in her post that she schools on the snaffle ring and that it's only hacking that she has it with any leverage action.
If you are already schooling on the snaffle and you find it is working well then I would recommend purchasing two bits - one for schooling in the arena and another for hacking. Find a snaffle that she really likes - personally I find that a lot of horses really like the double jointed snaffles (like the Sprenger KK, but you can get a cheaper version). My horse actually prefers a thinner bit, so you're going to have to find what works for her. This will be a great bit to school her in and get her foundation/training firmed up.
For hacking - I do understand that a lot of horses can get quite riled up out in the big world. Yes you do need to fix her training to an extent - but a more substantial bit is going to help you get your point across especially because the mare has a sensitive mouth. What I would suggest is getting away from a crude leverage bit, like a gag or kimberwicke, and instead purchase a bit with a solid mouth piece, port, shank and attachments for two reins. You should be able to find a suitable pelham that allows you to ride on snaffle action most of the time, but gives you a curb rein option when you need it.
A more pony-club style solution is to grab some baler twine and tie one end to the saddle and the other to the bit so if she does pull down, she wont be able to and soon she will learn that she can't root against you with her neck while out hacking. It's also called a grazing rein I think?? You just have to make sure you don't have it tied too tight so her neck is restricted in normal motion and it can't be too loose or it wont work.