I do have to agree with Tiny, that western riders do still want contact to some degree. We eventually move our finished horses to a curb, where they're riding with loose reins, but they started out ridden with contact in a snaffle.
Understand that, a well trained horse was started in a snaffle, and with pressure and release, learned to lift their back, and then their head followed their back.
I still, to this day, school my 15yo show horse, in a snaffle, on contact, every now and then just as a refresher on his self carriage.
What's your reason for the hackamore? I really don't think you will effectively achieve this without a bit.
To build a topline, you have to have a driving hind end, and that starts
with some form of contact. As they become more self carried, the contact can go away.
So, if there's not a medical reason for him not to have a bit, I think you should start there. You've gotten a lot of great advice on some exercises, as well.
You don't need an english saddle to school "english" things. Exercises are exercises, doesn't matter the actual discipline. I always will be doing very dressage-y schooling on my pleasure horse, western saddle and all.