Originally Posted by Nabterayl
I don't think the Inquisition has anything to do with this. A group of humans that are not a part of the Imperium doesn't really fall under the Inquisition's mission statement. That is a matter for the Ecclesiarchy, which is part of the reason I question the timing here. Virtually every starship in the Imperium, even rogue trader vessels, would have clergy onboard, and even the lowliest cleric would surely recognize the divine imperative to evangelize fellow humans who somehow have survived for thousands of years outside the light of the Emperor. Shipboard chaplain communicates the existence of the Republic to the Ecclesiarchy via the ship's astropaths, the Ecclesiarchy dispatches missionaries, the missionaries fail to achieve a 100% conversion rate, the Ecclesiarchy requests a crusade be mustered under its leadership, and the Ecclesiarchy-led crusade comes in its billions to spread the Imperial Creed.
That's really the only way I can see this happening. I'm not saying the Imperium needs to crush the Republic, just that the events you've described lead inexorably to the Imperium trying to crush the Republic, and so far there's no convincing explanation as to why that hasn't happened. Two thousand years is a really long time, even for Imperial bureaucracy, so I'm wondering why the Republic hasn't had to deal with an Imperial crusade yet.
In many ways this situation is similar to that of the Tau Empire, which has had to deal with an Imperial crusade, and was really only saved by the arrival of the tyranids. The key differences that you've identified so far are that the Republic makes contact with the Imperium way before the tyranids arrive, and heathen humans would probably be a higher priority for the Imperium than yet another minor xenos empire.