Thoroughbred schooling horse for advanced riders acts like jerk even in the indoor what do I do?

My instructor told me how wonderful he is but I don't think anyone rides him or has ridden him professionally in quite some time. My instructor wants him in a martingale. He's not young and Im sure he's had one on before. He won't go into the bit pressure and goes around like a prancing deer about to explode any second. My instructor tells me my hands should be in front of the saddle but I don't want to ride him with short reins since he's not listening. I've been on him twice and can get him to trot up with nice big strides and relaxed, its only when I start to shorten the reins that he goes stiff and looks for things to spook at. I've cantered him and it went amazingly well but he really hates it when I do a half halt and shakes his head. I think its because I do small half halts with just my fingers really quick and to him it felt like a jerk instead of a steady give and take. He's a tooth grinder. I've never ridden a horse that grinds the bit before. My instructor said he does it way more with the inexperienced riders and she was impressed that he hardly did it at all with me (probably because I had so little contact on the reins and like to ride with my hands low and close to my body). I think draw reins on the lunge line might help him. He goes in a happy mouth bit I think. Its a snaffle with cushiony stuff around it. Im not about to re-train this horse for free but any suggestions? He knows leg aids fine.