Is it permissible to adopt "In the name of Allah, the Most Gracious, the Most Merciful" as a daily litany (wird) recited a specific number of times, and have any prophetic hadiths been reported regarding this?

The prophetic hadiths do not state that "Bismillah al-Rahman al-Rahim" (In the name of Allah, the Most Gracious, the Most Merciful) is a regular litany (wird) that a Muslim should adhere to. Rather, it was legislated to be recited at the beginning of important actions, for blessings, seeking good fortune, and seeking assistance. Its appreciation is demonstrated by an action appropriate to the context, such as saying "Bismillah I read" when reading.

Making "Bismillah al-Rahman al-Rahim" a daily litany with a specific number of repetitions falls under innovation (bid'ah). Acts of worship are based on divine command and adherence (tawqif and ittiba'), not on innovation. Every action that is not commanded by the Sharia is rejected from its doer. The legislated supplications (du'a) and remembrances (adhkar) are the best and safest.