Unfortunately, the current Negotiation mini-game doesn't handle this very well. Thinking a little more about it, it may be possible to make the mini-game a little more generic, and allow some interaction like this.
Currently, the Negotiation mini-game models a barter - two sides haggling over a final price. This threat would be two sides trying to sway a third side. One possibility would be to use the main pieces - tracks for each leader from the faction - and a total support track. But instead of representing barter, the end total support would map to disposition (either gained or lost). The leaders could still be placed on tracks, such that moving them to a supportive side aids the Stronghold, while putting them on an unsupportive side aids the Autarch, etc.
It may be possible to make the "raise an issue / address an issue" actions more generic as well. There are already the "create an advantage / overcome an obstacle" actions present in the narrative scene and the Skirmish. These would map to issues. Then it becomes the basic aspect mechanics in play. The concept of magnitudes would be lost in Negotiation - but Fate core brings that back a little bit by having succeed with style create multiple free invokes (which makes the aspect essentially have a magnitude).
A third type of diplomatic interaction could occur in the game during a Conquest mission, when the heroes are trying to persuade a native population to join the Stronghold. This fits better the barter model - although maybe not in terms of trading this-for-that. Instead, it might just be a set of difficulties that the Stronghold must hit, the result being a net population which joins the Stronghold.
