Recent Neutron Scattering experiments by Richter et al. [1] show that the presence of coil-crystalline block copolymer (PE-PEP) micelles in a mixed alkane bath suppresses the crystallization out of solution of the long alkane component at low temperatures. Motivated by these experiments, we study theoretically the thermodynamics and kinetics of lamellar coil-crystalline block copolymer micelles in a bimodal solvent to better understand the factors determining the anti-precipitation action of coil-crystalline block copolymers. We assume an Alexander-de Gennes brush model for the strongly stretched corona chains and explicitly account for the polydispersity of the solvent chains. For the thermodynamic distribution of solvent chains in the corona, we find a predominance of short solvent chains to long solvent chains in the corona phase compared to the solvent bath, both with and without nematic interactions in the corona phase. We also calculate the rate of crystallization of the long solvent chains onto the micellar crystal core and find that the rate is sensitive to both brush and core parameters. In particular, we predict that to maximize the rate, both Efold/k TNA and χn need to be made as small as possible, where Efold is the folding energy of the crystal core chains, NA the number of statistical segments of the solvated corona chains and χn parameterises the strength of nematic interactions in the micellar corona. This leads to the surprising result that for fixed Efold, the rate of crystallization is increased when we increase the molecular weight of the corona blocks.