The seminal work on the idea of constitutional monarchy is A Short History of Man by Hans-Hermann Hoppe.
Constitutional monarchy is another step in the decline from natural order to modern democracy. Let's trace the decline down to that point then:
Government comes from natural elites. In medieval Europe, private courts, nobles, and indeed kings were sought by freemen for either protection or conflict-resolution. Since none of these were monopolistic, they rather closely resembled a private law society (the libertarian utopia), moreso than any other civilization in history.
Soon the kings secularized and consolidated power, either destroying or subjugating the institutions that stood in their way (the clergy, nobility, towns, etc.). Law and defense came to be produced monopolistically, which is what libertarians stand against by definition. Yet the degeneration is incomplete: the king is still the permanent, allodial owner of his realm (hence the time-preference argument you probably know by heart at this point), not to mention his monopoly is *informal*, so his authority can still be challenged and competed against. This institution may be more predatory than before, but what followed was much worse.
The constitution served to formalize and codify the king's monopoly. The fact that his authority was made the letter of the law, rather than the tenuous circumstance of competition as before, made it possible for him to tax and legislate much more.
The decline from here should be well-known to anyone, libertarian or not, who visits this board.
As to your suggestion in particular, it demonstrates a lack of understanding what constitutions are: namely, expressions of the ruling class' will. Hence, the absolute monarchs drafted absolutist constitutions to better secure their power and loot their subjects. For there to be a libertarian constitution, monarchic or not, there must be a libertarian elite, which does not currently exist and probably won't for the foreseeable future. It really is a long-shot to count on the circulation of elites for your political projects, so I would consider your idea a dead end.