Step 1: The key trick here is timing. The modern (NICHD) BPD definition first requires at least 28 days of supplemental oxygen, and then assigns severity strictly according to the level of respiratory help being given at 36 weeks postmenstrual age for babies under 32 weeks at birth.
Step 2: Map the support against the calendar. Birth was at 27 weeks. Add 4 weeks of ventilation to reach 31 weeks, then add the 1 week of oxygen to reach 32 weeks. From 32 weeks onward the infant breathes plain room air.
Step 3: At the decisive 36-week mark the child therefore needs no oxygen and no pressure support at all. Although ventilation appears in the criteria for the severe grade, that ventilation was over by 31 weeks, well before the assessment window.
Step 4: An infant who consumed at least 28 days of oxygen but is breathing room air at 36 weeks fits the mildest category. The need for more than 30% oxygen or positive pressure at 36 weeks (moderate or severe) is absent.
\[\boxed{\text{Mild BPD}}\]