Comment by Dra on Trash Of The Count’S Family - Chapter 126

Because the prince was heading to the place with sun church. Those guys hate the dark elves, who have and use dead mana as it's seen opposite of light. His disguise uses dark mana so going to the main location of where the sun church worshippers are had a huge risk of the light exposing his disguise and revealing he had dark elf blood. The bracelet originally was there to strenghten his disguise while he's practically surrounded by people who want dark elves dead. The blessing on the other hand from Cage strenghtened it further, making it more difficult for sun church's blessings and power to remove his disguise.

Without the bracelet, the prince would most likely have been executed after being busted for his disguise coming off and for having dark elf blood, even if it was just a quarter of the blood. It's all mentioned in previous chapters.