The sentence is
二人のやり取りに、当事者でありながら置いていかれた形の美智子は不満そうに唇を尖らせると.
I believe that であり may be being used here to connect the two verbs together but I'm not certain.
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Sign up to join this communityであり is the continuative form of である (a literary and stiff copula), so it's used to connect verbs in a broad sense. However, you have missed ながら, which in this context means "albeit/despite/although". 当事者でありながら means "although she is a person involved (in the matter)".
当事者でありながら置いていかれた形の美智子
Michiko, who has been left out of the conversation despite being an involved person