In this paper, we consider a class of DNA storage codes in which the selected molecules are constrained to be taken from an ``inner'' codebook (associated with the sequencing channel) which is used in a black-box manner. We derive the exact error exponent for this class of codes, which strictly improves over an existing achievable error exponent. Moreover, our achievability analysis is based on a coded-index strategy, implying that such strategies attain the highest error exponents within the broader class of codes that we consider.