How might Justice Joseph Bradley’s argument that a law barring women from practicing law could be upheld because of women’s “timidity and delicacy” be regarded as an application of the rational basis test? How might it be regarded as unreasonable?
Although the concept of the rational basis test post-dates Justice Bradley (making it rather suspect to try and analyze his position through that lens in the first place), the statement - as phrased in the OQ - does follow the organic root of the idea: that when discriminating against non-suspect classes, the government must have some kind of coherent reasoning leading them to that policy choice as preferential over others.
Bradley's argument, the "timidity and delicacy" of women, hinges on those attributes being held as fact. If this were factually the case, then one might presume women to be unable to satisfy the job requirement of being "zealous advocates" for their clients.
As for how it might be viewed as unreasonable, again the matter hinges on whether or not it is objectively true that women are naturally timid or delicate. The Rational Basis test does require that the Government produce SOME evidence that they've verified the factuality of an important claim. Even by the standards of the 1890s, the claim that women are naturally timid and/or delicate is dubious at best. By then, the Women's Suffrage movement had a full head of steam, being roughly 40 years old and nearing success. The descriptors being used to describe suffragettes of the day were anything but "timid" or "delicate."
That one could claim all women were naturally "timid and delicate" while at the same time decrying a lack of both of those of some women is attributable only to the human capacity for eliding cognitive dissonance, not any rational function.