Robert, it's recommended to hold bird-dog position to begin with. It is quite tricky exercise on the terms of the posture, people easily can get hurt if they go into motion without learning how to hold proper posture and be stable in it, specially for those who already have low back problems.

When something is weak, then something else will fire up to try to compensate. You then build other movements over those "bad" patterns and then it becomes really hard to break out of it unless you can find ways to, like you said, turn off the other muscles involved. How you go about this will depend on the person and the exercise. Having bands distracting your legs (forcing you to engage the hamstrings) will essentially turn off the hip flexors. The shape of your spine as a whole (top and bottom curve) will also vary which muscles will take over a certain position/movement (hence why you say you feel it more if you pulls your shoulders back, it probably affects your pelvic position too. Actually, just holding that "upright" position, if you have a forward head posture, is some work for the abdomen as well (you need the upward pull and stability on the front in order to get the downward pull in the back). PTs can look at you performing an exercise and think it looks good, but they can't feel what you feel. I have a wonky left shoulder and the PT gave me a rear delt exercises that ended up working my front delt more than the back because of my pulling mechanisms, and we had to try 3 different exercises to find one where I could actually isolate the back of my shoulder without the front taking over. So keep trying, and take notes of what you find really works your core, and discuss the fact that you don't feel certain exercises with your PT, they should have variations you can try that could work better for you!

Press your breath into your ribcage, causing ribcage expansion, via Foundation Training decompression breathing.