Congratulate the person who got the part you wanted. Being gracious costs nothing and becomes a reputation.

The moment right after losing something to a specific person is when your character is most visible, largely because it is when almost nobody manages anything gracious. What you do there gets noticed by the people making future decisions, and by the person who got it, who will be in your world for years.

Send it within a day, before it gets easier to say nothing. Keep it short and free of anything about yourself. Then go be disappointed, which is entirely reasonable, and let those two things sit separately rather than letting one contaminate the other.