Managed to pare my score down to 10 pieces (incidentally also worth 10 points in the usual piece value reckoning) by entirely locking down the position, thus removing any last remnants of fun the solver might have found in the earlier versions:
If white starts with
1. c3, black can thwart white's suicidal plans by promoting at b1, which causes an instant stalemate. White's only other legal move is
1. c4, and after that, there are no choices to make, so everything runs on railroad tracks until move 5, when white must underpromote to either a knight or a bishop in order to avoid checkmating black.
Earlier answer versions below.
Here's my attempt at constructing a position where underpromoting the 2nd rank pawn is the only way to self-mate in 5:
As you can see, I've split the problem into 3 parts: a prison for the white king, a prison for the black king, and a prison for the black rook.
The only way to force the black knight to move is to stop the black king from moving around in his prison.
The only way to do that is to promote the h-pawn to a rook and not a queen, as if white chooses a queen then black has ”5. Na3/Nd2” to avoid the checkmate.
Here's the same idea with 12 pieces only, this time requiring a bishop promotion.