Yes, it is possible to change your school in class 12th within the Karnataka Board.

Inform them about your intention to transfer and inquire about the Transfer Certificate (TC) process. Research schools you'd like to transfer to and see if they have seats available in Class 12th for your stream. Each school might have its own application process. Inquire about the documents they require and deadlines for submission. These documents might include your TC, mark sheets, and a character certificate from your current school.

https://school.careers360.com/boards/dpue-karnataka/karnataka-state-board-1st-puc-syllabus-bsar

I hope it helps!