People always resist taking a half breath in the upper register.
Do a stomach lift (without playing). Then you will understand why a half breath works and can use it quicker.
Take a full breath, pull the stomach as far in as possible and try to lift it toward the lungs.
You can’t do it.
Take a full breath and blow most of it away. Then try the lift. It is easier to do.
This way I can prove to you that a full breath PREVENTS the abdominal muscles from properly compressing the air for high notes.
The compression (bringing the stomach in) will speed up the air. The lift (pulling the stomach up to the chest) will force air from your lungs.
Practice half breaths and the stomach lift WITHOUT a horn. Just work on the breathing.
Use both the lift and timed (half) breathing together.
Be a breath centered player not a face centered one.