I think, but am not 100% sure, Microsoft restricts what can be on a CSV.
Of course, you can put the VHD there. That's the whole point. And the VHD could contain the database data and log files -- who knows what's "inside" a VHD.
But, not knowing your configuration, I don't know what you are trying to do. CSV is there so a single volume (e.g., a single LUN on a SAN) can be shared by multiple cluster members with individual files there in being used by different cluster members. Specifically, the information making up a VM definition and VHD.
Previously, one had to put the VM definition and VHD in a separate LUN so it could move about individually. There was nothing "wrong" with this other than the complexity of having so many LUNs.
Database files are different. You don't have as many. They are big. You want to carefully place them and watch them. Etc.
If you just put the database files inside a VHD then, as said originally, all is easy, except you don't get the detailed treatment you probably want.
If you put the database files in a separate LUN then you have all of the detailed treatment and that LUN will failover as easily as anything...