When Court of Vantress becomes a copy of a permanent, it's neither entering nor leaving the battlefield. Any enters-the-battlefield or leaves-the-battlefield abilities won't trigger.
Court of Vantress
The game starts with no monarch. As a player becomes the monarch, the current monarch (if any) ceases being the monarch. There is never more than one monarch at a time.
There are two inherent triggered abilities associated with being the monarch. These triggered abilities have no source and are controlled by the player who was the monarch at the time the abilities triggered. The full texts of these abilities are "At the beginning of the monarch's end step, that player draws a card" and "Whenever a creature deals combat damage to the monarch, its controller becomes the monarch."
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Nino Vecia
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
