Using a combination of small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM), we document the composition-dependent morphologies of 39 new poly(lactide-block-1,4-butadiene-block-lactide) (LBL) block polymers, comprising a broad dispersity B segment (M-n = 4.5-17.7 kg/mol; D = M-w/M-n = 1.72-1.88) and narrow dispersity L end blocks (M-n = 0.6-15.3 kg/mol; D = 1.10-1.21) with volume fractions 0.26 <= f(B) <= 0.95. A subset of these samples undergo melt self-assembly into cylindrical, lamellar, and apparently bicontinuous morphologies. By assessing the states of order and disorder in these triblock polymer melts using temperature-dependent SAXS, we find that broad B segment dispersity increases the minimum segregation strength chi N greater than or similar to 27 required for LBL triblock self-assembly relative to the self-consistent mean-field theory prediction chi N >= 17.9 for narrow dispersity analogues. While B segment dispersity has previously been shown to thermodynamically stabilize the self-assembled morphologies of low chi/high N ABA triblocks, the present study indicates that broad B block dispersity in related high chi/low N systems destabilizes the microphase-separated melt. These observations are rationalized in terms of recent theories that suggest that broad segmental dispersity substantially enhances fluctuation effects at low N, thus disfavoring melt segregation.