This happens because of django/django@23e319d it effectively means that TemplateView and JSONResponseMixin doesn't.. well.. mix.
I would suggest renaming content_type -> content_type_json, get_content_type -> get_content_type_json which would make it consistent with the names of the other methods. But that change would need a deprecation policy, so a design decision is needed. :)
For those of you that want to do this anyways, you can get away with this:
from braces.views import JSONResponseMixin as BracesJSONResponseMixin
content_type = None
Then just swap out your JSONResponseMixins.
I think your suggestion is a good one, but, you're right, it probably needs a deprecation policy. We'll see if it's possible to do it in a backwards-compatible way, and, if not, we'll mark things for deprecation starting in 1.4.0.