Hacham Eliyahu Shama Ha’levi (Chief Rabbi of Aleppo, d. 1814), in his work Machshireh Mila (12:44), addresses the case of a Mohel who performs a Berit on Shabbat for a child on the child’s eighth day, but is unable to perform the Mesisa – the extraction of blood from the wound after the circumcision. This can happen in the case of a Mohel with an infection in his mouth, and could thus endanger the child if he performs the Mesisa. The Shulhan Aruch rules that the two stages of the act of circumcision (the “Mila” and “Peri’a”) may not be divided among two different Mohalim when a Berit is performed on Shabbat, and the question thus becomes whether somebody else can perform the Mesisa if the Mohel performed the entire circumcision.
Hacham Eliyahu Shama writes that since it is considered dangerous to perform a Berit on an infant without Mesisa, this situation falls under the category of “Piku’ah Nefesh” – a potentially life-threatening condition – such that the Shabbat prohibitions are suspended. Even though extracting blood from a wound is normally forbidden on Shabbat, it is allowed in this circumstance, due to the potential risk for the child if Mesisa is not performed.
However, the Hacham adds that a Mohel who creates such a situation requires atonement. If a Mohel knows that he would be unable to perform Mesisa, he must not perform the Mila at all, and should instead have another Mohel perform the Berit in his stead.
Summary: If a Mohel has an infection in his mouth, such that he cannot perform the Mesisa after the Berit, he should not perform a Berit on Shabbat; a different Mohel should perform the Berit. If, however, the Mohel with an infection did perform a Berit on Shabbat, somebody else should perform the Mesisa.