QYBURN: The Unsullied have breached the gates of the Red Keep.
Maegor's Holdfast would be a better place to wait out the storm.
per the wiki:
Maegor's Holdfast is a massive square fortress inside the heart of the Red Keep in King's Landing. Named after King Maegor I Targaryen, it is a castle-within-a-castle and is the strongest place in the Red Keep.
The holdfast lies behind walls twelve feet thick. A dry moat lined with a bed of formidable iron spikes surrounds it, with a drawbridge spaning the moat. The only way in or out of Maegor's is across the drawbridge. A knight of the Kingsguard is always posted at its far end.
The royal apartments are located in Maegor's Holdfast. The king's bedchamber has twin hearths. The Queen's Ballroom within Maegor's Holdfast is much smaller than the Great Hall of the Red Keep or the Small Hall of the Tower of the Hand. However, the graceful ballroom can still seat one hundred.
As to whether it would have been any safer it's hard to say. Certainly it would be more resistant to normal invasion by opposing military forces but whether it could have withstood magical dragonfire is open to question.