Anything concerning freedom of religion for the time period in question is absolutely out of the question. The Catholic Church, up until Vatican II, and really not until the Pontificate of John Paul II, refused interfaith ecumenism and maintained that the Roman Catholic Church was the only true faith and all others were heathen. Even today, with Benedict's own special brand of ecumenism, many Catholic lay people are chaffing against what they see as anti-Catholic liberalism permeating the Church hierarchy. Even if a radically different Church view on the rights of the people and the role of government were to come about, tinkering with ideas concerning dogma is harder than making a diamond out of soap.
I could see various POD's that would allow a radically-republican Church supporting revolution in various nations, but then one has to ask the question:
Would the Holy See, in its capacity as direct ruler over the Papal States, support revolution in its own territory? Even a very pro-human rights Church today exercises complete and absolute control over Vatican City in the same mode it did when the Papal States were still in existence prior to the Unification of Italy.
I can see the Church supporting reform and constitutional government better than I can see the Church supporting revolution, especially in countries like France, Spain, and Austria. To support a revolution in these nations that had, for centuries, defended the Papacy against all comers, would be paramount to theological suicide. There would be major schisms all over Catholic Europe, especially in areas such as France where the Church was entirely way-too-intertwined in the government of the state.
France would not have been able to support the Church once the Church started sponsoring or supporting anti-monarchial revolutions, and so would have worked with French elements of the Church who were opposed to this sort of thing to get rid of the Papal elements of the government and replace them with anti-Papal elements in order to effectively schism without too much upheaval in the state and government. Put down a couple of minor insurrections and you have the "Most Catholic" Kingdom of France joining the ranks of the Protestants.
Or you just have France, Spain, and Austria electing an Anti-Pope and putting down the revolutions anyway.
I think the focus should be on a more reform-minded Church rather than a revolution-minded Church, though I like the idea of the Church putting everything on the line to condemn slavery in all its forms, though this would really only apply to the British and the Americans as the Spanish, French, and Portuguese colonial policies of inclusion pretty much butterflied away too heavy an involvement in slavery and made it more beneficial and economically prosperous to facilitate the slave trade until they all got stuck in Haiti and the French had to do something with them. In this instance, a Church that is staunchly anti-slavery would probably have succeeded in helping the French establish a department in Haiti, or relocating many of the former slaves to French possessions in Africa.