My concern is the 'no Christian sect not having preference over another'. Of course I believe that the Catholic faith is supreme to all, and should be given preference. That's why I, more and more, am finding that republicanism is incoherent with Catholicism. I know Pope Pius XII said otherwise, but I humbly disagree with him.
I can think of some republican constitutions that are compatible with the Faith, such as the aristocratic republics of Venice and Genoa, for example, or that of Switzerland and Florence. Indeed, that is a true republic, akin to that of Rome and the Greek city-states, also described by Aristotle as the politeia (polity) or later described as a mixed régime (regimum commixtum) in Scholastic political philosophy. As far as I understand it, the Faith is compatible with any form of government that is not contrary to the moral law, as was affirmed by Leo XIII, Pius XI, and Pius XII. Furthermore, Leo XIII established that "the people" are not sovereign but rather those who elect the executive merely select the one who God then bestows with sovereign authority. This power can be shared or rather diffused, however, to legislators and judges, or at least that is what I gather from various XXth-century theologians who commented on the matter.
Nevertheless, monarchy is the most perfect of all governments, and the monarchy that maintains strong elements of aristocracy and republic within its constitution (while remaining a true monarchy, with a truly sovereign King who exercises real power) does indeed seem to qualify as a mixed régime. This sort of monarchy, per St Thomas, is the ideal, and it existed for many centuries in the Mediterranean and later in what became known as Europe within the past several millennia, both prior to and after its conversion to Christianity. I would say that this kind of non-bureacuratic government by men for the common good is the most stable, as its longevity testifies. The thing that seems to have destroyed it is the competition between the nations and the rise of the money-power, a twin and symbiotic process that dimmed and obscured the undying vision of a united Christendom and a restored Roman Empire to establish a true peace in the world.
The general rules of thumb, then, according to my thinking, are the following :
I. Governments that do not respect the inequality between different social stations, which is part of the natural order, are monstrous (in the Aristotelian sense of being unnatural) and abhorrent.
Note : To paraphrase Leo XIII, "nobody does not profit from this natural inequality between the classes" unless they agitate against it due to some misguided principle and thus convert a source of harmony and order into an excuse for chaos.
II. Republics and such like are tolerable if less than ideal (and they have a unique tendency towards persecution of the Church insofar as they might treat clerics like lay subjects of the state), but they typically only work in smaller polities; large countries require either a monarchical principle to unite them.
Note : Otherwise, they can only be united through a tyrannical, massive bureaucracy that more so dominates through the petty violence of an infinity of regulations and standardisation processes than it rules. And this point, I say, history proves.
III. Some governments are detestable and are irreconcilable with the Faith, such as tyranny, democracy, oligarchy, or something contrary to the moral law.
Note : An example of a government contrary to the moral law would be one wherein only women are allowed to rule or a state based on the ruler's ownership of all his subjects and their property as chattel slaves or a state wherein membership in an evil society or an evil oath is required for enjoyment of the franchise. A state wherein one must be a sodomite or must commit murder to enjoy be enfranchised would be an example of a State with which the Church cannot have dealings.
One could argue in some cases that the lattermost requirement might be a simple law rather than a feature of the constitution as such and that therefore it is subject to removal. Perhaps so. After all, the Church does not deny the right of pagans or infidels to rule their various states; the Church grants that non-Catholic princes truly enjoy sovereign authority provided by God. So, it seems that a state that could be said to be opposed to the moral law has to be so at the natural level rather than the divine, otherwise no non-Catholic state would have the right to exist.
The other problem is that the liberal republics tend to be evil and one must swear an oath to the principles of liberalism in order to exercise government office in most of them. The degree to which this compromises the government's reconcilability with the Faith would need to be evaluated on a case-by-case basis, but I personally believe that it poses a substantial compromise. One might argue that the Church has tolerated and spoken well of such republics before, but then again the Church also initiated diplomatic relations with the Soviety Union, or rather the Vatican City-State did so under Pius XI on the specious grounds that an opportunity for the advancement of the Faith was opened upon the removal of the Tsar, "our greatest enemy." Pope Pius VI also officiated at the auto-coronation of Napoléon Bonaparte, but that did not mean that the small corporal was a legitimate monarch, as most Catholics and many Catholic bishops in France long maintained and still maintain without censure.
I think it bears stating that the political alliances and choices of the Holy See are not binding on Catholics' consciences. The Church can excommunicate Kings and can remove their subjects from their obligations of obedience, but this is only an indirect power rather than a direct one, and the Church long ago dropped the discourse of plenitudo potestatis that would give the Holy See the ability to make or break any state by fiat. Even holy Popes like Innocent XI gave support to William of Orange against James II of England -- a Catholic who would have arrested the English Protestant oligarchy and propagated the Faith in Britain -- in order to thwart the designs of Louis XIV during his most Gallican and regalist phase. Yet, it would be absurd to say that Catholics were and are required to praise the so-called "Glorious Revolution."
There are perhaps some other general guidelines that could be thought up, but I am tired, my brain is shutting off, and I do not want to continue writing this post.
Anyways, I don't think someone could say that this amendment is not better than our current constitution. But still it still places our Holy Faith on the same level with false religions, could a Catholic Man, in good conscience, support it?
I personally would not be comfortable supporting it and would reject it. I think balkanisation would be preferable to a constitution that officially declares the Catholic Church to be treated as an equal constituent with Protestant sects and as a subsidiary organisation simply participating in the life of a precedent city. The Church is sovereign and is due by divine right the prestige and favour of the laws. Even so, the Church has tolerated certain abuses of her authority before, such as in the Concordat of Bologna with François I of France and similar concordats with other powers wherein the Kings of Europe would have the right of nominating candidates for the episcopate (which more or less meant indirectly electing them), or in her concordats by which she tolerates the non-recognition of her rights in laic or religiously hostile democratic republics in the contemporary period.