When you decide that transgender people aren't really fully members of their chosen genders you're denying them exactly that.
Also, I have to say, it sounds to me as though some wires are getting crossed in terms of the word bigotry here. A lot of people seem to be under the impression that the only thing which qualifies as bigotry is direct out and out hatred and violence. What people need to understand is that within a social justice and a wider anti-oppression context, the term bigotry refers to anything which serves to marginalize, dismiss, or otherwise harm or oppress disadvantaged groups, or anything which serves to perpetuate social systems or social norms which already have this effect. You may ask what good the term is if it can refer to such a wide range of behaviours. The point is that if we limit the term bigotry to only the very worst forms of hatred and violence, it becomes all to easy for people to congratulate themselves on not being a bigot, when literally all you're doing is the bare minimum; not actively and consciously participating in a behaviour which is obviously oppressive, hateful and violent to minorities. If things are going to improve for marginalized groups people have to hold their behaviour to a higher standard than that, and so if we can make it clear that actions and words which are oppressive in more subtle ways can also be harmful, this becomes a lot easier to do.