You should continue your birth control pills. Having on and off with a BC will make your hormones dizzy and not be able to ovulate very well. If you consumed more than 4 pads per day (fully soaked) or more than 1 pad per hour, We might consider it as bleeding. Your doctor is right, you should double dose but in the time that it is already bleeding, you have to seek for another medical appointment. and to add, make sure when you use pills, you are not pregnant and if you missed one, take two pills the following day. If you missed again and have a sexual contact, better have a preg. test or better yet, seek for medical help. :)
What you are having is a side effect of the birth control pills as heavy bleeding is a frequent side effect. I would advise a gynecologist’s evaluation and getting drugs like oral contrecptives and tranexemic acid prescribed to arrest the bleeding.
I hope it helps. It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.