Jcecil3 has accepted my invitation to start a Catholic political party. He's named it the Catholic Peace and Justice Party, which should give you some idea of its platform, if you have any experience with "Peace and Justice" Catholics.
As a draft platform, it clearly shows the problem with creating a political party "good on all aspects of Catholic social teaching." You can be unassailable on moral principles, but as I wrote below political policies are derived from other kinds of principles as well. If your economic principles are unsound, your policies based on them will be unsound. The Church tries not to pronounce on the soundness of economic principles (except insofar as they have moral implications), so two Catholics can legitimately accept mutually contradictory economic principles, and therefore legitimately support mutually contradictory policies. (Steven Riddle points this out using the example of gun control.)
Here is my hypothesis: Most American Catholics have their opinions regarding politics and government formed with little or no reference to what the Church has to say about politics and government. Politically liberal American Catholics tend to think government ought to do more than the Church says government ought to do. Politically conservative American Catholics tend to think government ought to do less than the Church says government ought to do. The former need to guard against a quasi-Pelagian belief that man can, through wise government, establish a natural paradise. The latter need to guard against a quasi-Calvinistic belief that citizens are predestined to a certain level of success or failure.
I think we could use a coherent, Catholic political philosophy. ("Catholic" in the sense that the philosophy is informed by Catholic teaching on faith and morals, and so neither too rosy nor too bleak, rather than the sense that it is the Official Political Philosophy of the Roman Catholic Church.) The scraps of political philosophy I've come across come across as having whatever is "Catholic" about them forced into a pre-existing framework.