I've always wondered why everyone assumes the Caldari and Amarr are Best Friends when their alliance is nothing more than mutual interests and realpolitik aimed at containing the Federation diplomatically and territorially to their own advantage. Just because State Megas do business in the Empire does not mean they automatically condone its culture and politics, it's just the nature of corporations to seek their expansion into as many markets as possible for their own (And the State's) interests.
Kaalakiota for one has stations in the Federation, Empire and Republic and potentially continues some degree of operations in all three under the SCC. It does this because it wants market share and profit like any corporation, if it didn't expand operations and penetrate markets then it wouldn't be a business. Kaalakiota, like most Caldari megas is solely interested in advancing their own power and agendas and this extends generally to the State because that mindset of there being only ones own interests pursued at the expense of everyone else lies at the core of the Megacorporate mindset through the CEP and also held in varying degrees by its citizens.
The State and the Megacorporations could generally be said to be concerned solely about relative power compared to the rest of New Eden and particularly in ensuring its national survival through maintaining economic and military strength in comparison to others. With that worldview, the greatest threats to the national survival of the State are the Federation and Empire, and when the NAP with the Empire was signed the Federation was still very much a legitimate threat to the State if not an outright hostile entity whose motives were to be treated with suspicion.
Even with the change of relations between the Federation and State after Yioul and the formation of CONCORD where there was a re-engagement of sorts in bilateral relation and cross-border trade brought a thawing of relations, an alliance with the Empire still remained a relationship of strategic importance because it was insurance against hawks and nationalists in the Fed acting upon their agenda by having a powerful nation nominally aligned against them right on their border in addition to granting Caldari Megas preferential treatment over their Federal counterparts in Imperial markets.
The reason the State does not pursue an alliance with the Republic is that it simply does not grant the same strategic and economic benefits that its alliance with the Empire does. The Republic might be viewed by the State as essentially being a failed state on Federal life-support and taxpayer funding whose tribal system leads to government inefficiency and corruption that's just bad for business and whose military is not significant enough to hold off both the Federation and Empire in any future conflict in which the State and Republic are allies. It might be said that slavery should be cause enough for a State-Republic alliance but then again it's not Caldari citizens that are being held in chains; the Minmatar were more than glad to enslave each other in internecine wars in the past; and frankly, any Caldari leader that places supposed ethical considerations before the interests of the State and their corporation would be guilty of gross incompetence and more for the only moral imperative is to ensure the protection of the interests and maintenance of relative power of the Caldari State and people.
The only real relationship I see between the State and Republic is one of maintaining a phony war across their mutual border without actually invading and making assurances to that effect in private while seeking to play the Empire and Federation against each other so that they are able manipulate both in their own interests of gaining funds and materiel while ensuring a movement of opposing military units elsewhere to secure their own war fronts for themselves against their larger adversaries.
Given that view I'm not really sure where the whole Caldari-Minmatar outreach comes from because I don't think even the Liberals or Ishukone concerned themselves with the issue of Minmatar slavery when they sold TCMC to Khanid/Amarrian Holders to trial on slaves or that Gariushi's crisis of conscience over Insorum and giving it to the Minmatar only arrived near the end of his life. Unless it's Caldari playing the dissident Civire angle from the background, who move to the Republic for the same reasons as the Brothers of Freedom due to disagreements over the system in the Caldari State, yet still believe themselves patriotic enough not to live in the Federation during their exile with their diaspora brethren.
In the end, if the Republic is unable to provide tangible benefits to the State both economically or strategically then any discussions about formal alliances really is a moot point because for the citizens of the State the world is divided between those who are Caldari and serve State interests and then there's everyone else - you serve your own interests before worrying about the problems of others who are not citizens. Some might find such a worldview distasteful, but every nation in Eve does the same to one extent or the other otherwise they'd be consigned to the dustbins of history. The only difference is that the Caldari State is honest about it and doesn't wrap up its intentions in democratic and religious platitudes or moral and ethical self-aggrandizement.