Coordinated Attack Problem Different Requirements
This looks like the Byzantine generals problem.  There's a lot written on that subject; I suggest you start by reading the available literature and material on that subject.

I'm not entirely clear on you mean by "solvable in system S" (system S says processes don't fail, but the weak termination condition refers to failures, so what's going on there?), so I can't confidently answer that part of your question.

There's an obvious algorithm for the case of two parties:


Each party continually broadcasts a message saying "my input is X" (fill in X with whatever its input bit is).
Whenever you receive one of those messages, you respond with an acknowledgement describing your input, what you received from the other party, and what you will output; then you immediately produce the appropriate output and halt.
If you receive an acknowledgement message and have not halted yet, then produce the appropriate output and halt.


It looks like this satisfies all three of your requirements, including weak termination, assuming that messages can be dropped/lost but are not modified in transit and no forged/fake messages are inserted by the network.