I have to say I don't see a real three-way contest. C's objective is divorced from those of the other two and seems to have no opposition. C's involvement can be handled mostly narratively unless one or both of the others decide to oppose C's objective.
So, it works. C either assists one or the other or both or neither.
Or joins one side or another.
All of these can be handled. It is probably a good idea to give C AP
in case he/she joins in, but it may not be needed.
C doesn't lose APs unless targeted by an exchange.
If C wants to make C's presence a condition of B going I would first ask if B opposes. If B does not then C can simply join B. Then we can ask if A wants to accept that B can go if C goes with B - if A accepts then A can simply join C.
If both A and B are opposed, then we get to the complex situation.
After some thought I would make that a three-way contest.
If B goes to zero then C has the option of continuing the contest against A.
if A goes to zero then C can continue the contest against B.
In either case I would probably allow the knocked-out character to provide one Augment to C, because it makes sense in the narrative logic (although I might require C to have won an exchange against the knocked-out character).
If C goes to zero first then C is just out of the contest altogether.
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