S.W.
Our responses to hormone changes are SO different, that one woman's response to perimenopause/BCP can be completely opposite another's.
I was on the pill up to age 51 and had started having symptoms that seemed like perimenopause for only about 6 months. My doctor recommended I go off the Pill, and my symptoms disappeared!! I have no more hot flashes, night sweats, digestive difficulties, anxiety, etc. etc. I'm still having regular monthly periods now at 53.
My sister, on the other hand, went off the pill, had no periods, and was told by her doctor that she was completely through menopause, blood tested and everything, when she was 50 and she'd had NO symptoms. THEN she started having all the symptoms after she stopped the pill.
You and your doctor will need to strategize about what is happening, and what to try. You may want to go back to a regular 3 wk/1 wk pill cycle, or try a different pill, or a different BC method, to see if that helps.