(A.) Whoever desires to achieve something should sacrifice

It is easy for an objector to go on and argue as follows:

A Śūdra (i.e., a member of the lowest class) desires to achieve something
A Śūdra should sacrifice (PMS 6.1.25)

Mīmāṃsā authors, however, reply:

No, because sacrifice presupposes knowledge of the Vedic prescriptions enjoining it, such as (A.), and a Śūdra is not entitled to hear the Veda. In fact, there is the following prohibition:

(B.) A Śūdra should not engage with the Veda

(Śābarabhāṣya ad PMS 6.1.37)

Now, why is the prohibition (B.) stronger than the prescription (A.)? I can think of two or three possibilities:

1. Because (B.) is more specific than (A.). That specific rules overrule more generic ones is known as the upasaṃhāranyāya.
2. Because prohibitions have a bigger deontic value than prescriptions.
3. (Because of sociological reasons: Śūdra could not be allowed to sacrifice because there was a social consensus about the fact that they were not allowed to perform sacrifices)

The last explanation is easy, but I am afraid it might be too easy. Mīmāṃsakas were not directly involved with worldly matters and could engage in brave thought experiments, such as asking whether animals are entitled to sacrifice. No. 1 is fine and probably applies here, although one needs to be aware that multiple rules may act simultaneously, so that, e.g., in the case of the Śyena sacrifice the same rule is not enough to overrule the generic prohibition to perform violence in case of the malefic sacrifice Śyena.

For more on the Śyena, see this post. For more on Mīmāṃsā deontics in general, see these ones.