Is it obligatory, recommended, or legislated for one who awakens at night and recites the prescribed remembrance to then invoke blessings upon the Prophet (peace and blessings be upon him) and supplicate for whatever he wishes?

It is narrated in Sahih al-Bukhari and other collections, on the authority of 'Ubada ibn al-Samit, that the Prophet, peace and blessings be upon him, said: "Whoever awakens at night and says: 'None has the right to be worshipped except Allah, alone, without partner. To Him belongs all sovereignty and all praise, and He is over all things competent. All praise is due to Allah, and glory be to Allah, and there is no god but Allah, and Allah is the greatest, and there is no might and no power except with Allah,' then says: 'O Allah, forgive me,' or supplicates, his supplication will be answered. And if he performs ablution and prays, his prayer will be accepted." Supplication here is recommended, and it may be obligatory in certain cases, such as supplication in Surat al-Fatiha during prayer, or in the funeral prayer, or in the Friday sermon according to some jurists. As for sending blessings upon the Prophet, peace and blessings be upon him, it is a Sunnah, but it is among the reasons for the acceptance of supplication and is required before it.