The correct answers are be sleeping, not awaken, and should be sleeping.

Adjectives expressing importance or necessity (like crucial, vital, essential, imperative) also trigger the subjunctive mood in the following that-clause.

• be sleeping: This is the continuous (or progressive) subjunctive, formed with be + present participle (-ing).

• not awaken: This is the standard negative subjunctive (not + base form).

• should be sleeping: This is the "putative should," a perfectly valid alternative to the subjunctive, widely used in both American and British English.

Does not awaken uses the indicative mood, which is inappropriate for formal subjunctive structures, and to sleep is an infinitive, which cannot act as the main verb in a that-clause.