Is it permissible to write "صلى الله عليه وسلم" (may Allah bless him and grant him peace) when a passage or message about invoking blessings upon the Prophet (صلى الله عليه وسلم) is mentioned, or is it sufficient to say it silently?

Invoking blessings upon the Prophet when he is mentioned is legislated, either as an obligation or a recommendation. It is better to write it along with uttering it. If you suffice with uttering it, it is sufficient. Imam Ahmad did this, and he avoided symbolizing it or omitting from it.