What should the brother do to document his marriage and his wife's conversion to Islam?

Your brother's marriage to this woman, according to Christian faith, is invalid because it contradicts Islamic Sharia (definition). The validity of an Islamic marriage requires the presence of two just Muslim witnesses, among other conditions not met in Christian marriages. There is no impediment to him marrying her again after she has completed her 'iddah (definition) (waiting period) with one menstrual cycle. Since she has embraced Islam, only a Muslim man can be her guardian (wali) for marriage. If there is a Muslim man among her family, he has the most right; otherwise, any Muslim man can serve as her guardian.