[Ethernautica]

Hypothetically, how could I add a second capacitor boost stage to this circuit to brightly flash blue/white LEDs?

Circuit is from the Talking Electronics website

 

[Delta Prime]

Hypothetically you can't.
That is a high gain positive feedback, regenerative circuit,who's output voltage maximum would be 2.4 volts.
A jewel thief or blocking oscillator, boost converter circuit topology would be preferable. White and blue LEDs take More voltage to turn On.

Typical Forward Voltages by Color
• Red: 1.6V – 2.2V (typical ~1.8V – 2.0V)
• Orange / Yellow: 2.0V – 2.2V
• Green (Standard): 2.0V – 2.4V
• Green (True / High-Brightness): 3.0V – 3.5V
• Blue: 2.5V – 3.7V (typical ~3.2V – 3.6V)
• Violet / Purple: 2.76V – 4.0V
• White: 2.7V – 4.0V (often uses a blue/UV base with a phosphor layer)
• Ultraviolet (UV): 2.1V – 3.8V