Real Buddhists have no problem being recognized as a part of the wider Dharmic philosophy and we don't mind this. The concept of religion is a European/Islamic invader trend that has nothing to do with the thousands of years of tradition of debates and philosophy that we collectively have within Buddhist/Hindu society.
You are confusing the Ambedkarite logic with Buddhist philosophy, which is completely wrong. Ambedkarism is a political religion that is a mirror of the Abrahamic totalitarian mindset. The same goes for the Westoid mentality. Just as they classify Catholicism and Orthodoxy as different religions despite them both being Christianty, they also distinguish you from us.
It is not something that's going to change anytime soon, so instead of explaining it to them and dissing us, you might actually be better off telling them to suck it.