The Didaché clearly delineates key Catholic rituals being practiced as early as the 2nd century.

Rituals which became associated with Catholicism. It was not Roman Catholic-specific at that time. There was no central Catholic organization. The Church hierarchy was fragmented.

What documents prove that the the early Christians weren't Catholic?

Nope. You're shifting the burden of proof so that I'll have to prove a negative in order to answer that. That's like if I asked "what documents prove that Jesus didn't travel to America?" Obviously there aren't any documents which prove that he didn't because no document addresses that. In other words, there's an absence of evidence altogether. Now before you try to use the following argument as a justification, I concede that absence of evidence is not necessarily an evidence of absence, but that argument is constantly abused by Christians, and to apply it here in favor of your position proves nothing (see "Russell's teapot" for an illustration). The burden of proof is on the claimant. Your claims about Catholicism are traditional stories which the church used to ex post facto rationalize its legitimacy. Such traditions are not evidence. Without legitimate evidence contemporary to that time period, there is no evidence for the theological supremacy of Catholicism.

We do, however, have evidence for decidedly non-Catholic hierarchies- the second century piece The Shepherd of Hermas, for example, depicts 2nd century authority as belonging to a confederation of elders- no bishops or central authority.

In the Bible Jesus clearly states that he would establish the Church with Peter as the foundation

There's this revisionist history among the Catholic hierarchy that Catholicism was established and given credibility by Jesus himself. But really, that's an extreme over-interpretation of Jesus' ministry and his communication with Simon Peter. He doesn't say "the Church". There's no definite article there and no capitalization of "Church". That's really just a sly way to try and portray Catholicism as being the only entity to result from this exchange, so that Catholicism as an organization can portray itself as the Church. But the reality is that Jesus' church existed as multiple loose organizations prior to the centralization of the Church- according to Eamon Duffy, Christians felt that they were part of a central body (the Body of Christ), but not a central organization. Additionally, we have multiple legitimate regional churches (later diocese, although still distinct from Rome) which later became the Oriental and Eastern Churches. These began breaking from the rest when the Pentarchy began asserting their authority over Christianity.

All in all, independent organizations have just as legitimate a claim to Jesus' "my church" reference as anyone else, because there's no evidence that Jesus' commandment to Simon Peter referenced a particular future organization. This becomes even harder to stomach when we consider that multiple organizations schismed off the decentralized results of Peter's work- Peter and Paul's spread of Christianity didn't just result in one lineage.

Not that I care. I'm an atheist.