No. We really don't.
If there is a fork, then one or the other of the two chains dies within a day. Whatever is left is BTC.
If it works out like this, then I think everyone agrees that the remaining chain can be named "Bitcoin". However, I don't see how anyone could know for sure that it will, indeed, play out this way. As far as I can tell, we never had a fork that was even comparatively as disputed as this one.