I've searched the cyber version of "A transcript of the registers of the Worshipful Company of Stationers, from 1640-1708, A.D." without success.

Francis Grove is mentioned on 10 Jan. 1648/49 and on 11 Oct. 1653 only.

Neither case bears any obvious relevance to this discussion.

As a matter of interest, Mennes's "Musarum Delitiae [sic]" is entered at 1 June 1655.

I find no mention in the 1640-1708 Registers of any publication called "The Miller and the King's Daughter" or "The Miller's Melody," or any at all with "Miller" or "King's Daughter" in the title.

The same holds for the Registers of 1554-1640.

Presumably James Smith is credited as the author because he wrote or rewrote the piece as it appeared in 1658.