Stabilization is a good goal. Since I only get to ride the horse for 30 minutes a week I just slip in stabilization as a rest/reward when I start stabilization. Walking on a loose rein requires stronger neck muscles for effective gaits. Since you ride an ASB I presume his neck is thin and longer than the norm, and it may take him a little longer than you want for him to get strong enough to do a good stabilized trot full of impulse and under control.
Maybe you could try just walking for several rides. I work on the three speeds of the walk, doing turns in the three speeds of the walk, turns in place, fore and hind, and backing up.
I work to get the horse’s back “swinging” freely and feeling the push of the hind leg through my seat. I keep contact, giving all the rein the horse needs to stay comfortable for this more challenging work at the walk. My contact is measured in grams, not ounces or pounds, and when I walk when my hands move forward I relax my fingers and I emphasize the most forward position of the horse’s head by opening my fingers maybe an 1/8th of an inch before gently closing as the horse’s head starts moving back. When I feel the push of the hind leg I relax that finger, so I end up alternating the hand with which I do the extra 1/8" or so.
I now think that riders often dismiss the walk. I find that by working as above at the walk that the horse gets stronger and more able to easily obey me at the trot. It is NEVER a waste of time to work at the walk. I mean work, not shuffling around off rein,working at the walk means walking on contact, using alternating legs, and adapting to the movement that the horse willingly gives me.
As a break for the horse I, off contact at first, practice my two-point at the walk, still moving with the horse. Only when I do not feel like I need to grab the mane do I start keeping contact.
When the horse does all the above easily and with a powerful drive from the hind legs, then the trot will probably be easier. Your horse still remembers what you were working on, now he has to get used to your new “language” and strengthen the muscles that he needs for the more Forward movement.
Enjoy!