The Rama (Rav Moshe Isserles of Cracow, 1530-1572), in discussing the laws of Berit Mila (Yoreh De’a 264; listen to audio recording for precise citation), writes that when choosing a Mohel to perform a Berit, one should search for a Mohel who is not only knowledgeable and proficient, but also righteous. The Lebush (Rav Mordechai Yoffe, c. 1530-1612) explains that the Mohel’s intentions when performing a Berit can have a profound influence upon the child, and therefore it is preferable to choose a righteous Mohel whose intentions at the time of the Berit can positively impact the infant.
Interestingly, however, the work Machshireh Mila (4:18) cites the Shulhan Gavoah (Rav Yosef Molcho of Salonika, 1692-1768) as advancing a different view. The Shulhan Gavoah notes a famous passage in the Midrash stating that the prophet Eliyahu refuses to attend a Berit Mila until Hashem forgives all the sins of everybody in attendance. As such, the Shulhan Gavoah writes, it can be assumed that all the people present at the Berit – including the Mohel, of course – have had their sins forgiven so that Eliyahu can attend. It thus turns out that everyone present at the Berit is, for those moments, a righteous person. Accordingly, there is no reason to search specifically for a righteous Mohel – because everyone present at the Berit is righteous. The Shulhan Gavoah observes that the common practice is, indeed, not to make a point of selecting an especially righteous person to perform a Berit. Moreover, he adds, every Jew has a “Hazaka” (presumed status) of being upstanding and observant. For this reason, too, it is not necessary to specifically choose a particularly righteous individual to serve as the Mohel for one’s child.
Practically speaking, then, there is value to try to find an especially righteous Sadik to perform a Berit, in accordance with the view of the Lebush, and certainly, one should not invite a Mohel who has a bad reputation. However, if somebody invited a Mohel with an acceptable reputation to perform the Berit, and he then hears of a different Mohel with a reputation for being especially pious, it would inappropriate to disinvite the first Mohel. Our Rabbis speak very harshly about one who breaks a verbal agreement, and this would apply in this case, as well. Therefore, especially in light of the position of the Shulhan Gavoah, one should not cancel the hiring of a Mohel in favor of another Mohel reputed to be more righteous. Perhaps, if one appoints a Mohel and then hears that one of the leading sages of the generation is going to be in the neighborhood and is available to perform the Berit, we might consider allowing cancelling the first Mohel, though even this is far from clear. But certainly, it would be improper to cancel a Mohel to choose a Mohel who is reputed to be more pious.
Summary: It is preferable to choose as a Mohel for one’s child somebody reputed to be especially righteous, though any proficient Mohel may be chosen, as long as he has a generally positive reputation. If one chose a Mohel and then hears of a different Mohel who is known to be especially pious, it would be improper to cancel the first Mohel.