I was in a similar situation once- a theatre asked me to be the dance captain. We got a special concession from AEA, and I have some (limited) dance experience, so we went with it. I wouldn't do it again. There was nothing terrible about the situation, but I found that I really missed having a second set of eyes to watch the dance sections, and a second voice in maintaining the dance. It was a show with a whole host of other issues, and it would have been better to have another person thinking about the dances in addition to myself.
In my understanding, with the selection of fight captain (or any other similar job that requires additional payment), it's the producer's decision, usually with some input from the fight director, who to hire. If this other actor is not suited to being the FC, the producer should decide who they want to hire. Personally, I think it helps to have a second person keeping an eye on the fight. It's good to have someone else concentrating on the fights (in addition to yourself) and someone you can have an ongoing dialogue with about what's happening and what maintenance notes need to be given.
You can make sure that the actor selected is always present when the fights are happening. (I would put his name on the call for that scene, as FC, even if he's not acting in it.) It's important to have someone as FC that is capable of running fight call, and understands the fights well enough to maintain them and the actors' safety. If this actor can't do that, they shouldn't be FC.