The board game I would SHL-ify is Secret Hitler, except it would be Secret Hamilton.

In the game of Secret Hitler, you have Liberals, Fascists, and Hitler.  The Liberals win by passing five Liberal Policies or killing Hitler.  The Fascists win by passing six Fascist Policies, or if Hitler is elected Chancellor plus three Fascist Policies.  The Fascists have to act in secret so they are not found out while trying to push their agenda.

In the SHL version, we have HO, GM, and Hamilton.  HO wins by passing five good rule changes or firing the Hamilton GM.  The GMs win by passing six rules with loop holes, or if the Hamilton GM is elected to Commissioner plus three loop holes.  The GMs have to act in secret so they are not found out while trying to game the system.

Throughout the game, there are two rotating roles President and Chancellor.  In our version, we'll call that Voice of HO and Commissioner.  Every turn, the Voice of HO candidate role rotates to the next player and they propose a Commissioner, then the rest of the players vote to elect that government or not.  When elected, the Voice of HO and Commissioner work together to pass a rule change from three candidate rule changes.  Loop Holes can grant one time special powers, like a Thunderdome that allows the Voice of HO to learn the true colors of the individual they TD, or a Firing where the fired player is removed from the game.  If Hamilton is fired, HO wins.