Theoretically, the Blues could trade Miller's rights after the draft (or in an unlikely scenario, sign him and trade him), owing Buffalo a 2nd round pick. The Blues could then collect that 2nd rounder from the team they trade Miller's rights to, and it would get them out of trade pick debt.
The question is, would any team trade a 2nd round pick for Miller's rights? Unless it's Garth Snow, probably not. And then still probably not.