Dumb question here - Do we really want to get involved in Kislev Civil war?
We might not have choice, but do we actually want to?
Honestly? Outside of picking up our daughter midway? I see no benefit to getting involved. Let's think of this on both sides, our wife's sister who a Druchii torture master supreme compared her rein to a dreadlord, which holy shit. Or whatever opposite side is opposing her, so let's say the church, the ones that are not ice witch and widow, the ones with a bear for a god ursun and other gods. Also the nobles.
We don't have the best view of the conflict, not the basis, the players, the reasonings, etc. so let's just look at it from a beneficial side. What has katterin done for us and can we expect her continued rein to be a positive one? I don't think it will provide any benefits to us or the empire, as I just don't think it's sustainable. Yes she has marched with us, but it has been very clear it was more out of duty and honor than any actual want. I don't think she will exactly be friendly to us in the future.
Ok so now the church/nobles. The actual benefits of their rein could be guessed at, but the detriments to their rein are very clear. They do not like foreigners/prefer an isolation policy, they like slavery and will undoubtedly start up the trade again as soon as they are in power, while they will undoubtedly still trade with us, there is little to any chance that our emperor will be able to continue his talks with forming a military alliance.
I lack enough information to list more, but these are the obvious ones. They have plenty of negatives, but outside of making any further inroads and military alliances very difficult, it has no effect on us compared to how they are now. The only difference would be the slave trade will pick up, which yes is bad, but it's not the worst thing in the world. At least compared to katterin deteriorating mental state. Seriously I think she's about to snap soon.
In other words, while I don't think we should support the opposition, I don't think we should support katterin either. Possibly just speed run finding our daughter and leave as fast as possible. It probably won't go that smoothly but that's what I think we should aim for.