Hard to say if constitutional monarchy could be made useful in game terms. In one sense, it is "transitional" government between monarchy and democracy (which suggest monarchy and republic as prerequisites). On possibility is that gets the trade bonuses of republic/democracy; doesn't suffer from the unhappiness of military units in the filed; doesn't have the population growth bonus of "We Love the King Day"; and the "senate" can force cease fires/peace treaties upon the King, but not as often as with Republic/Democracy.
Two other possible governments:
Social democracy: No unhappiness except that caused by military units in the field; these military units in the field cause even more unhappiness than under a democracy; senate can force cease fire/peace treaties even more often than under a democracy; "We Love the King" day allows cities the trade bonus as if it were a Republic/Democracy. ...
"Green" Socialism: Granted, we have to see truly green government yet; but in alternate historical timeline, it could be possible. Like "social democracy," military units in the field cause unhappiness even more so than democracy; likewise, pollution causes unhappiness. The principle advantage of this government would be that, through extensive environmental legislation, the civilization could produce more shields without pollution; but such environmental controls would decrease the amount of money generated by the economy. Furthermore, Solar Plants would turn content citizens into happy citizens in addition to their anti-pollution benefits; while prior to the discovery of fusion power, nuclear plants would cause unhappiness. (I have no idea what the benefits of "We Love the King Day" might be.) In general, a very pacifist government; but increased production without pollution would allow the building of city improvements more quickly.