I just checked, and doubled flame and doubled finale do NOT increase Doubler score. In a testing run, I only activated Doubler twice in the song - once I made sure the only note I hit was a flame note, and once I activated it after the last note of the song but before finale. My final doubler subcategory score was 0. The doubled flame note went entirely into the flame note subcategory, and the doubled finale went entirely into the Finale category.
It's really ****ing frustrating then to get so many points as a result of doubler only to find this doesn't contribute towards your doubler subscore at all, and you're effectively wasting a doubler activation for the purposes of this goal to try to, you know, maximize your doubled score with it.