I'm going to simplify this, because you've got your terms confused-- republicanism and federalism are two very different things and the two are completely compatible with one another.

Republicanism, especially in the context of the early United States, is the idea that the people (rather than a king or emperor) are sovereign, that just governments derive their powers from the consent of the people governed, and that governments exist to protect life, liberty, and the pursuit of happiness are what make for a free people. None of these basic tenets were questioned by anyone in the newly independent United States at the time the Federalist Papers were written.

The more important question was how best to protect the republic that had been established by the American Revolution. At the time, there were two major fears. First, it was clear that the Articles of Confederation, which had functioned as the constitution since 1776, didn't give the central government enough power. (When an armed insurrection against the Massachusetts state government started in 1786, neither the Massachusetts militia nor the U.S. government could muster the authority to put it down. Ultimately, Boston merchants ended up having to raise a private militia to put down Shays' Rebellion.)

But there was also another fear, as voiced by Patrick Henry and others. A strong central government was what the colonists had revolted against during the Revolutionary War, and the fear was that a strong central government would ultimately just trade King George III for another tyrant.

Federalism was the solution that balanced these two considerations in a way that everyone could put up with. Federalism provides for a strong central government, but places legally enforceable limits on what the central government can and cannot do-- the rest is up to the states. So, for instance, national defense, interstate commerce, and printing money are the exclusive responsibility of the federal government, but those powers are all explicitly authorized by the Constitution that Hamilton was advocating for.

At the time of the ratification controversy, the terms "Federalist" and "Republican" referred to political principles, not political parties. A "federalist" government was one that was a federation of smaller states, like the Constitution laid out; a "republican" government was one that was based on the rule of the people, like the Constitution also laid out. So, Hamilton and Madison could use both terms in the Federalist Papers. There weren't any organized political parties at the time with single names. The faction in favor of the Constitution is, and was at the time, normally described as "federalists" because how much the central government would be "federal" (stronger) versus "confederate" (looser) was a legitimate point of disagreement; all sides agreed that republican government was a good thing.

To confuse matters somewhat, when the first American political parties developed in the mid-1790's, they called themselves "Federalist" and "Republican." The Federalists wanted to position themselves as the ones who truly supported the Constitution, and as they had the stronger organization under President Washington and Hamilton, they got first pick of the name (as it were). The Republicans then adopted their name, in part from the "Republican Societies" that had sprung up in support of the French Revolution, and in part because they positioned themselves as the party truly supporting the people against the Federalists' alleged elitism.

If you want one book on the development of the early American party system, I'd recommend Bernard Weisberger's America Afire. Lynne Cheney's biography James Madison: A Life Reconsidered is also rather good (at least, the first half; I haven't finished the second half yet.)

Thank you all so much! I realized that the Constitution did support both a federalist and republican government; but as Lawyer1234 mentioned, I was thinking the two were exclusive from each other, when in actuality they do not have to be. All my confusion was stemming from my belief that a government could not encompass both federalist and republican principles at the same time. I appreciate all of your help!