Cob size bridles are smaller than standard size bridles. A cob is an English term for a smaller than average horse (but not a pony).
A side pull is to train a horse to yield and bend. Usually the young horse is first trained in a side pull and just wears the bit and gradually is brought to understand the bit.
A hackamore should be a rawhide core bosal with a mecate. When used correctly the bosal fits the horse's nose and face and the mecate actually squeezes the bars of the bosal so that the bosal works on more than the nose of the horse. The proper use of the rawhide core bosal and mecate is part of the Spanish Vaquero's horse's education toward the proper use and wearing of the spade bit.
I think you are referring the a "mechanical Hackamore" or 'bitless curb' that actually acts on the horse's facial structure similar to a levered vise. I have used a side pull and brought a horse along in a bosal and mecate but I have never used a mechanical hackamore and never would.
It sounds like your horse (and maybe you) needs a good deal of training. I suggest you use a side pull and have the horse wear the bit (snaffle) and gradually change him over to the bit (take weeks to do this). I used to use two sets of reins when I did start using the bit.. at first with no pressure on the bit while using the side pull. Eventually I would use both with equal pressure and eventually the bit rein would be doing the work, and the side pull rein would be slack.
There is something about the outside of a horse that is good for the inside of a man. ~Winston Churchill
) Dinosaur Horse Trainer