This is a Mullen mouthpiece. Is this the mouthpiece on your curb bit?
I myself don't have a problem two-handing a horse in a solid mouthpiece **if** you are schooling the horse and they know what you are asking. When I took my horse to a reining trainer earlier this year, they used both a correction bit on him (loose pieces on the mouth piece) and a low port reining bit (solid mouthpiece). 90% of my riding is neck reining and legs, but if he doesn't soften up or bend as much for what I'm doing, I'll pick up on my inside rein and ask him to soften and bend.
So if you want to ride two-handed ALL of the time, then you outta use a snaffle (no shanks) bit on your horse because that is what they are designed for.
Agree with Skyseternalangel. A lot of people think they are making a horse "soft" by putting pressure on the reins and squeezing with their legs, but that's the wrong way to do it. You can't make a horse soft in the body when you are only working on their face. You have to make the horse's body soft FIRST, and collected, and then the head will naturally drop on its own.
What type of snaffle you have in her mouth shouldn't matter, although yes one might work a *little* better for her than another, just due to the horse's preference. Maybe a double-jointed mouthpiece would work better. I personally hate bits that only have one joint; I feel like they nutcracker in my horse's mouth. I get much better responses with double-joints.
Go back to the basics with your horse. You shouldn't even think about loping right now, because you probably haven't mastered it at the walk. You say she can "bend at the poll" at the walk, but from how you described you ask her for it (pull on reins, and squeeze her sides) you have taught her the wrong thing.
THIS is precisely why working with a trainer (even just lessons once in a while; not even sending the horse there for 30 days or anything) can help immensely. You've been unintentionally making your horse's mouth harder by giving her confusing signals (go back .... wait go foward ...... wait, I don't know!)
Do circles, circles, and more circles. Circles creates bend in your horse's body and will encourage her to step her inside leg underneath her, round her back, and then causing her head to come down and relax. Use your outside rein with very light contact just for support. With your inside rein, you want to twist your wrist, and lift upward toward your opposite shoulder. This asks the nose to bend inward and the shoulder stay elevated. At the same time, put your inside leg on her in the middle, and ask her ribcage to move out, so that her body is bending around your inside leg. You can use your outside leg to keep her hip inward, if need be.
When she gets that, remove tension. Chances are she won't hold her body position and will go back to being straight. That's okay. Ask her to bend again. WHen she's doing it nicely, release pressure. She might not hold it. That's okay. Ask again. Eventually, your end goal is for her to self-hold in that bent frame for you.
Make sure to do this both ways. And don't leave that walk until she's holding herself. THen you can try the trot.
Notice NONE of that involves pulling on the reins and squeezing her to go forward at the same time............... That's an artificial head seat and artificial head drop. You can't just change one part of the horse's body. You have to change the body, and then the rest will fall into place.