To gain stability back when you lose it in the tack will require compensations to occur. The only way to regain stability on a horse is to use your core, there will be no negative impact the horse, meaning you will not shut the horses energy down.
Most horseback riders tend to regain their balance using their reins. Not using your hands for balancing on a horse is required for an independent seat.
The bottom line is a horseback rider will have an easier time developing their balance off the horse rather than on one. A rider should maintain their own balance in their core to keep them in the saddle. We are on moving animals and finding stability is needed to do other things in the saddle.
Teach yourself how to have an independent seat while staying stable using internal balance. Below is a sequence of exercises using a stability ball that triggers the balance similar to being in the saddle. I like the way the thigh lines up under the hip, developing a longer leg. Hip flexors must lengthen to a have the straight thigh which helps muscles work evenly around the hips and lower core. This balance is dependent upon in the saddle and the same muscles that help you balance on the ball are helping you balance on the horse.
When a rider has developed proper stability in their bodies, they can feel if and when they are not in balance.
Start 30 seconds
All Fours on Stability Ball
Then move to
Kneeling on ball for 30 Seconds