It is very difficult to lose the right to appeal in FED. Firstly, FED allows verbal dead ball appeals, and if time is requested in order to make an dead ball appeal, the umpire must grant it (presumably once action has relaxed).
So it would be pretty uncommon to see a relaxed action live ball appeal. There is a risk and no gain by doing so.
Caseplay 2-29-6B shows that even if the pitcher, during relaxed action, throws the ball into a dead ball area during a live ball appeal, the defense hasn't lost its right to appeal, provided the pitcher announced the appeal. (The runners do get 2 bases.)
If a live ball appeal is attempted, and the offense successfully initiates a play on a runner, the right to appeal isn't lost.
However, to get to your question, the appeal has to be attempted before the next pitch or illegal pitch. I'm not sure if a balk is necessarily an illegal pitch, but many balks are also illegal pitches. Anyway, an illegal pitch costs the defense the right to appeal.
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