Solution:
Hello Romain and welcome to StackOverflow. It looks that the new theme that you have activated did conflict with your plugin “wp-booking-calendar” this usually happens when 2 plugins or a plugin and a theme use the name for 2 functions or class. IN your case its a class name.
The error says “Cannot declare class wp_booking_calendar_public_setting,” this means that on that page something else is using the same class name “wp_booking_calendar_public_setting” usually plugin can not get such conflict because they are using their custom names.
I can help you if you shared with me the theme full name, I will dig in its code and get he issue.