Why is DMSO more polar than other solvents, yet it dissolves ionic compounds worse?
Having large dipole moments and dielectric constants, as DMSO does, does not imply that it has the ability to solvate ionic compounds like $\ce{CuCl2}$.  

Although both methanol and DMSO have oxygen atoms with their lone pair of electrons to help solvate cations, there is no corresponding functionality in DMSO to solvate a small, inorganic anion like chloride.  Methanol on the other hand can solvate chloride via hydrogen bonding in the same way that water does.  

These properties are not necessarily applicable to a coordination compound, so there is no reason to expect methanol to act as a better solvent than DMSO in this case.