How did the Prophet (peace be upon him) approve Bilal’s addition to the Adhan, “Prayer is better than sleep,” knowing that acts of worship are prescriptive? And is it proven that the Prophet (peace be upon him) approved the Companions’ additions to any other act of worship?

The Hadith you mentioned, in which Bilal (may Allah be pleased with him) added, "Prayer is better than sleep," was authenticated by Al-Albani. There are narrations indicating that the Prophet (peace be upon him) was the one who commanded Bilal to do so. The Hadith does not suggest that Bilal added something to the Adhan without a legal basis from the Lawgiver; rather, he said it in response to someone who informed him that the Prophet (peace be upon him) was asleep. Then, he commenced the dawn Adhan by the Prophet's command. The religion has been perfected, and it is not permissible for anyone to add to it or subtract from it. Legislation is what the Prophet (peace be upon him) approved, not the personal interpretations of individuals, for every innovation is a heresy.