"Berserk king" is my own invention ;). It means that you play according to the normal chess rules, until a king is mated. At that point the mated king is allowed to do TWO moves, in a last attempt to save his life. The first move does not have to be legal of course: as per the definition of mate, he will still be in check after the first move. And its second move may capture the enemy king which will even win the game for the berserk king!
The question now is: make a position that can clearly be achieved with the normal rules and then show how you can mate the black king from that position such that you win even if the king goes berserk. During those last moves no pawns may promote or pieces be taken.
For example, the following position won't be accepted:
[FEN "7k/4K3/8/6RQ/8/8/8/8 w - - 0 1"]
because there is no way to reach this position with optimal counter play of black. Playing backwards from this position must uncheck black, and give him space that he could have come from. There is no such backwards move that will force black to go stand in the corner however.
Likewise, the following position - although berserk-mate is not acceptable:
[FEN "7k/4KRQ1/8/8/8/8/8/8 w - - 0 1"]
because it would only be reachable if the white queen took a black piece on its last move.
Finally, the follow isn't correct either:
[FEN "6k1/5R2/5K2/4R3/8/4N3/8/8 w - - 1 0"] 1. Nf5 Kh8 2. Re8#
because, although black can't escape check in two moves; they can capture the white king and win! Not berserk-mate thus.
The question is, what position can you come up with that has the least number of points on the board, from which you can reach a berserk-mate without capturing pieces or promoting pawns? That is, Queen=9, Rook=5, Knight=3, Bishop=3 and pawn=1 point.
The best that I could come up with so far (although I didn't try very hard: it has to remain an achievable challenge to improve it ;) is:
[FEN "5B2/4KP1k/8/8/8/8/8/6R1 w - - 0 1"] 1. Rg2 Kh8 2. Rg6 Kh7 3. Rf6 Kh8 4. Rh6#
at 9 points, where no two moves of the black king can bring it out of check or capture the white king.
As a fun side-note - if the position doesn't have to be reachable (but it should be in order for it to be a correct "answer" to this question) then the following would be a solution with just 3 points:
This is mostly a reaction to @DialFrost 's comment, asking if the position has to be "reachable", at which I stated that it could be an interesting question to ask for any position, as long as white can force berserk-mate in at least -say- 3 moves; without that white is allowed to promote (which would be a higher number of points imho).
Here is what I came up with, for that case: berserk-mate in 6 with 3 points :P
[FEN "1b1b1b1b/b1b1b1b1/1b1bKb1b/b1b1b1b1/1bBbkbpp/b1b1b1bp/1p1p1p1b/1n1n1nb1 w - - 1 1"] 1. Ba6 Kf3 2. Kf5 Kg2 3. Ke4 Kh1 4. Bd3 Kg2 5. Be2 Kh1 6. Bf3+