Thanks to @JohnnyK for posting this code, I was able to write the following function to set all of the outputs to use 3.3V:
void set_all_pins_to_use_3v3(void)
{
DigitalOut enable3V3Pin(P3_6, 1); // Connect 3V3 to VDDIOH
volatile uint32_t * const iomanUseVddiohRegister0 = (volatile uint32_t *) 0x40000D00; // Set the register location
*iomanUseVddiohRegister0 = ~((uint32_t) 0); // Turn all bits on
}