Sounds to me like you just need a good coach who knows how to ride a horse's hind end up to the bridle.
Your post makes it very clear to me that you are riding front to back, trying to pull his head in to give him a 'pretty headset' rather than working his body correctly in order to achieve a round and fluid outline.
Forget about pulling back and wiggling your horse. There is little wonder that horse goes better in a bitless if you are going this - his mouth isn't being made sore and uncomfortable.
Put him in a snaffle - either a loose ring single jointed, or loose ring french link. If it's a single joint, make sure he has 1cm on either side of his mouth to allow for the 'shortening' action of the bit. If a french link, it can be a little tighter fitting.
Invest in a neck strap (a stirrup leather will do just fine), and for your whole ride, hook your fingers under the strap. Your hands are not to move - especially backwards.
Think of it this way:
Pulling back on the reins pushes the hind legs out behind.
Pushing the hind legs forward into the reins, pushes the neck and wither forward into the bit.
You don't want ANY pulling action while riding, everything is pushing.
So, keeping your hands still, resting on the wither and hooked under the neck strap, start your ride, and don't think of ANYTHING other than making the hind legs active. Ride millions of transitions, changes of rein, circles, serpentines, leg yield etc.
It will take time for your horse to consistently come 'onto the bit', because he has lost trust in your ability to give him a soft and consistent contact to work into. But it will come eventually, as you persist he will learn to trust that you're not going to start pulling on his mouth and winging his head around.
Your absolute best bet here, is to get lessons with a Dressage instructor, or an eventing instructor that has a good background of Dressage (assuming you are riding English).
You can put all the bits in the world into his mouth, and strap all the pieces of leather onto his head to pull it down, but you will never, NEVER have him working softly and correctly into the bridle if you don't learn how to ride it.