It is probably balanced
There is actually precedence for this in the variant rule for Lingering Injuries on page 272 in the Dungeon Master's Guide:
Horrible Scar. You are disfigured to the extent that the wound can’t be easily concealed. You have disadvantage on Charisma (Persuasion) checks and advantage on Charisma (Intimidation) checks. Magical healing of 6th level or higher, such as heal and regenerate, removes the scar.
These injuries are intended to provide either a neutral or detrimental effect, so the game designers, at least, see it at as balanced.
In my experience, ability checks that involve Intimidation are prompted by the GM much less frequently than other Charisma checks, but that will vary from table to table and from game to game. Overall, I don't see a balance issue with your proposition.
All that being said, it is ultimately up to your GM whether to allow it in the game, as there are no rules to cover starting with injuries in the rulebooks (and the Lingering Injuries rules are optional).