The Neji/Aika divide is going to be quite difficult to sort out a good resolution to.
Supposing for a moment that we do not have the ability to get both on board (and note that we would also have to make sure Neji, and Hinata by extension as Hyuuga Clan Head, are okay with this), then we have to choose which one to include and which one to exclude.
If we side with one of them over the other, then we distance from the other by the implicit snub. It would be somewhat worse on this axis to cut out Neji, since we approached him first and held no reservations about cutting him in. But the question of who we would offend more is counterbalanced by the relative cost of a snub to either side. Do we care more about our relations with the Minami than we do our relations with the Hyuuga? Is it more important to push the Hyuuga further away from hostile relations than it is to keep the Minami happy?
We could also cut both off, delivering only a minor snub to both clans in that we were not willing to take their side, but also reducing the size of our network.
The fact of the matter is that we've stepped into the mire of Minami/Hyuuga hostility and our actions here will be seen as our stance on the conflict. It won't be game-deciding, but if we side with the Hyuuga here the Minami would worry that that's the start of a general trend instead of a single decision. Neutral positions are also possible, but they need clear work to that effect if we want to make that stance. This also ties into how we want the Goketsu's political reputation to be. We're a strongly liberal clan, but are we aiming to push our side to victory or maintain peaceable relations between us and those we disagree with?
Personally, I think we don't have enough power to try and push our side of politics to victory, and so I don't want to take the routes that alienate the Hyuuga. If we can't alienate the Hyuuga, we can either alienate the Minami (not good but not terrible), cut both out and aim for neutrality, or go for broke and try to get both of them to cooperate.
If we try that last one, we should make sure it's able to fail gracefully. Also, maybe Mari has some ideas on what stance would be best for the Goketsu to take.