NP hardness of unique Puzzle Generation
Your problem is in NP.
Generalized Sudoku is NP-Complete, but your mapper is in P. Check whether given grid is generated by mapper runs in $\mathcal O(n^2)$ time.
Your mapper is not NP-Complete and your decision problem is not NP-hard.

On the other hand decision version of Sudoku is NP-hard.

The main problem is that you have not provided anything about checking uniqeness of solution, but since your mapper resolves only one solution from family of solutions it can't decide it.

If you insist on proving NP-hardness please provide valid reduction (hint, it will not run in polytime, so answer is no).