ISL85403 buck-boost converter
The 1st problem is the high Ron resistance of Q1 = BS138 = 3.5 Ohms.
  This ought to be ~10 mOhm ~100 mOhm . The IC BOM list  BSZ100N06LS3G is 10mOhm is used in order to boost the voltage by sinking high current from the inductor then "flying back" that current into a boost voltage during off time.