The reasoning by Medhātithi on 8.316 (on punishment and why it does not violate the prohibition to perform any violence) is quite complex and multiple opponents discuss. I will list them below attributing a number to each one of them.

1. A first opinion being discussed is that, since punishment is enjoined (presumably: as part of the duty to protect people) and prohibited (by the prohibition to inflict any violence), it is optional. “Option” (or “free picking”, vikalpa) is in general to be avoided and Mīmāṃsā authors would accept it only for details bearing no great significance. Hence, the opinion is dismissed.

2. Next, a second speaker observes that no vikalpa applies, because violent punishment is clearly forbidden by the prohibition to perform any violence.

3. Rebuttal: It is not forbidden because it is performed as part of fulfilling one’s duty.

2a. Second speaker again: in order for the prohibition to perform violence not to apply, the act of (violent) punishment should be enjoined, as it happens in the case of the Agnīṣomīya (a sacrifice which involves violence, and which is not blocked by the prohibition to inflict violence).

3a. Violent punishment is enjoined through worldly experience, hence no prohibition affects it.

2b.The second speaker then expands on the above point and observes that there is no specific prescription about violent punishment, since punishment is only derived from the duty to protect people, but this duty could be fulfilled also through other means (e.g., reprimanding people). Hence, inflicting violent punishment is not enjoined as a duty and it is just a worldly act (and, as such, liable to be blocked by the prohibition). Similarly, if one were to perform a sacrifice leading to a violent result because one desires it, as it happens in the case of the Śyena, that sacrifice would be blocked by the prohibition to perform violence.

2c. The second speaker also belabours on the difference between instrumental and resultative violence, saying that in the case of the śyena violence is prompted by desire, whereas in the case of the Agnīṣomīya, it is merely subsidiary.

2d. If violent punishment were specifically prescribed, then we would have to resort to vikalpa (as discussed above).

Thus, the conclusion appears to be that violent punishment should not be resorted to as a consequence of the duty to protect, although (as discussed in my article on corporal punishment) it can be resorted to in order to purify the culprit.