I don't think a martingale or tie-down would work...yes it keeps the head down, but it can harm the horse, and give them a bad impression about you, about you riding her, about accepting the bit/bridle.
I've never really known a horse to toss his head simply because "he was lazy". Not comfortable with something something, yes....in pain, yes, but not because they are lazy.
Try just giving voice commands. Make sure you bring up YOUR energy when you ask for the trot. If you feel dead on her back, it won't "encourage" her to move forward. Try to move WITH the horse.
My horse is trained that the minute my body stops moving with him, he is top stop...your horse may be green but who knows...maybe she's thinks that way also.