However, does the rest of the Alliance have sufficient cause to interfere with an "internal D'Bringi affair"?
That's basically my question: to what extent do the 'core' Alliance governments have an obligation not to interfere with the internal affairs of the others?
The Alliance didn't directly intervene when the Torqual government decided to murder and exile a large portion of its citizens, although IIRC they did provide indirect support to the rebels.
However the Torqual were only associate members, not full members, and IIRC this was before any integration started, so I assume that different prohibitions and obligations apply.
The Alliance, while apparently powerful when viewed from the outside, is a careful balancing act between the interests of its core powers. All had something to gain from the formation of the Alliance, but none wanted to give up any internal autonomy. Obviously, to make something like a multi-racial federation work, they'd have to give up some, but the treaties that created the Alliance gave it numerous powers to deal with inter-species relations and foreign situations, but almost no powers to regulate or interfere with the internal issues of its member states. Given the immense power of the Alliance, it could do almost whatever it wants, regardless of the desires of a single member race, or what the laws or its creating documents actually say. However, since its start the Alliance and its ruling council have bent over backwards to create a custom of not interfering in the internal affairs of member races, which is why they were so conflicted and limited in dealing with the Torqual. Because the Torqual were an associate race, the Alliance Council could have easily come up with a rational for supporting whomever they wanted, but, directly interfering in the Torqual situation would have created a precedent for meddling and direct military intervention. This would have had two direct results - first, it would have created an almost overwhelming desire to do it again, given the power available to the Alliance Council, and secondly, it would have created great distrust amongst the member races, as each began to fear that it would be next. The Alliance Council rightly feared that taking action in the Torqual situation would eventually lead to the destruction of the Alliance through fear and mistrust.
The new D'Bringi Emperor just performed an extrajudicial execution of an important Alliance individual, on the equivalent of live TV. On the one hand, if he was worried about the foreign policy implications he probably wouldn't have broadcast it to the other governments.
On the other hand, because they are allied and somewhat integrated, it seems likely that they have some obligation to protect an allied incumbent government against violent internal threats. I think it would be very easy to apply that classification to this new proto-government, and therefore be under an obligation to intervene rather than a non-interference obligation.
If a strong non-interference obligation does exist, I can think of 2 ways around that.
First, the execution and murder of various high-ranking D'Bringi and the interference with the Alliance ICN (and related systems like the Rheorish naval intelligence databases) constitutes an attack against the Alliance. This would justify removing the D'Bringi from the Alliance, and consequently remove any obligations to support the new government or refrain from interfering in internal affairs.
Secondly the Rheorish and T'Pau could determine that the D'Bringi Empire is not a legitimate successor to the old D'Bringi government, and therefore the Alliance would have an obligation to support the remains of the old government against the Imperial upstarts.
Of course, none of this matters if the Rheorish and T'Pau decide to take a see-no-evil approach.
The new D'Bringi emperor fears exactly this. He was confident that his coup would work, and that the general population, which he believes longs for the old days, would support him. His biggest worry was intervention by the other Alliance powers. Thus the misdirection by the ICN, and the raids on the Colonial Union. The D'Bringi have a long tradition of raiding, and as a traditionalist, the new emperor doesn't see them as an act of war. He understood that the humans would be upset, and would demand repayment, and would threaten war to get the reparations, but in his world-view raids are a legitimate tool of diplomacy and cultural expression, and he does not understand how the Colonial Union will view them. The new emperor knew he would need time to consolidate his rule to become the sole choice for the Alliance to represent the D'Bringi, and temporarily crippling the ICN and provoking the humans were intended to get him that time. Being focused almost entirely on internal matters, he may be failing to understand how the other races are taking this.
The Rehorish and the T'Pau are coming to understand that all of this is a result of the actions of the D'Bringi Empire, and when the human problem explodes into their awareness, they will immediately tie that in with the ICN issues and be able to blame the D'Bringi. They are critically aware of the vulnerabilities of the Alliance now that communications are compromised, and, if pushed far enough, will decide that the D'Bringi have become a threat to the Alliance.
The saving grace for the Alliance in all of this is that none of their enemies or competitors know this is happening. Even the associate members of the Alliance have little idea of the disruption within the Alliance.
If it comes down to war with the rest of the Alliance, the D'Bringi are in a completely compromised position. As a race that prefers ST type planets, their colonies are scattered through the Alliance, surrounded by the more numerous colonies of the Rehorish and the T'Pau. The Alliance has colonized 28 type ST planets, compared with 141 type T planets. And, although the D'Bringi plotters were able to begin moving their ships before the rest of the Alliance races, this was in aid of their misdirection plan, not in preparation for fighting a civil war. Indeed, most of the crews and officers of the various D'Bringi crewed ships don't even know about the plot, just that the government back home has changed. That means they really haven't been asked to choose a side yet. If the emperor manages to consolidate power and eliminate rivals, then they will be presented with a done deal and have no real choice. If forced to choose between the Alliance and their race, most of them would choose their race, probably. They haven't been asked that question yet, though. For now, most of the D'Bringi officers are taking a wait-and-see attitude.
Even worse for the D'Bringi, their home world is isolated in a dead-end system, with the T'Pau standing guard over their one access point with the rest of the Alliance. If the D'Bringi had been planning for civil war, that is where they would have concentrated their strength.
Still, while the D'Bringi aren't really in a good position to fight a war, neither are the other two members of the Alliance. While they could undoubtedly beat the D'Bringi, what then. Conquering them and forcing them to remain members of the Alliance while they are rehabilitated is more of the old-D'Bringi way of doing things, rather than the more 'enlightened' Alliance approach. Letting them leave the Alliance is problematic as well, given how interpenetrated their territories are. Letting them remain as they are, after causing all of these problems is problematic as well.