So I do not want to use choice or HBT. If there were an example of a normed vector space with two inequivalent norms (may as well assume that $\|-\|_a$ is strictly weaker than $\|-\|_b$) so that the space is a Banach space with respect to the weaker norm then method (1) would work, but that would contradict the open mapping theorem (unless there is some subtlety that I can not see right now).