It could be that the saddle angle is forcing you back against the cantle. True, you need to sit up straighter, but if the saddle is too high in front (and it looks kind of like that to me), you will find it really hard to sit up correctly vertical, and when cantering will be consistently left behind in the motion.
Take and post a photo of the saddle on Byrdie with no one on board. Also, a shot of how you normally sit when you ride bareback. Your postition in the saddle would hopefully be pretty close to the position on bareback, tho not exactly the same.
Also,, try riding IN the saddle but WITHOUT stirrups , trot and canter.
If the saddle angle is not level (and this is quite possible with a sway backed horse) you might be able to lift it a little with a wedge pad, but it's kind of dicey because such lifting can then cause the saddle to press down too hard on the area just behind the shoulder blade.
After you take the above mentioned photos, try saddleing up with the saddle a tiny bit further back, just for giggles.
She looks like a really sweet mare. I envy you your skill at bareback. I am a total wimp when it comes to that.