Those are not valid sentences.
Removing する from a [N + する] verb has impacts on not only its object but also its subject and adverbs that modify it.
Since the first sentence has no subject, the object 任務 gets marked with の to form the noun phrase 任務の達成. あす and 必ず pose some challenge. If left as they are, they would modify the only verb left in the sentence, namely 約束する. If they are to modify 達成, they would have to be converted into noun-modifying forms. あす could be simply changed to あすの. However, since 必ず has no corresponding adjective form, it would have to be substituted with an adjective with a similar meaning, such as 確実な.
あすの確実な任務の達成を約束する。
The second sentence has both the subject (私たち) and the object (この薬). While not totally unacceptable, 私たちのこの薬の試用 would sound a bit awkward. This problem could be avoided by marking 私たち with に so it collocates with 認める.
彼らは私たちにこの薬の試用を認めた。