Is roaming free in St. Barths with my French or European (EU) SIM card?

No! This is the #1 roaming trap in the Caribbean. Although St. Barths is a French collectivity, it is NOT part of the European Union. Your EU/French "Roam Like at Home" plan does not apply. Most carriers (including Orange FR) will charge you extremely high international roaming fees, often the most expensive rates they have. A Cellesim eSIM is the only way to get a fixed, prepaid data price and avoid a bill shock.