This may not help, but when I got my first horse my mom just had me get on bareback and learn that way. Just riding bareback any chance you get (I would recommend ten minutes a day starting out very slow) will improve your balance.
If you can ride bareback I promise you can ride in any kind of saddle with or without stirrups. Just make sure to practice proper technique. Even though you don't have stirrups, keep your heels down and a lot of people tend to slump. Make sure to sit up straight like you would in a saddle! I promise your balance will improve and you and your horse's relationship will grow stronger.
I prefer bareback and my friends trainer says I have better jump position bareback than with a saddle. And yes, you can hold on to their mane for balance. Just a thought? You may have already tried this but I just thought I would throw it out there. Hope I helped some!
Weavin' and Leavin'
"Don't worry, be happy"