Probably I have missed relevant material; but I have heard the Archbishop of Canterbury, the Archbishop of York (ex-Staggers) and the former Bishop of London, and others, and I have not noticed any suggestion that we should pray for Prince Philip that, although mortal and fallen, he may be given rest among the Saints.
Praying for the departed has an interesting history in Separated Anglicanism. The government suppressed and looted the chantries in 1548 (the first really big intrusion of the Tudor Regime into parochial life). But in the 1549 Prayer Book the Memento etiam of the Canon was actually made longer. However, the 1552 Book cut out Prayer for the Departed (except in as far as it might be implied in phrases such as "all thy whole Church"). And every time liturgical revision has occurred in the Church of England, the Evangelicals have struggled to make sure that (1) Eucharistic Sacrifice; and (2) Prayer for the Dead, are kept well out.
It has been suggested that the Saints crept back into the C of E through stained glass windows; Prayer for the Departed certainy made its reentry through Church Music,
W J Birkbeck (1869-1916), Fellow of Magdalen College in this University and a great admirer of Slavic Orthodoxy, did a translation of "The Russian Contakion for the Depparted", which appeared in the 1903 English Hymnal. It has achieved great popularity in elite and educated Anglicanism because of its haunting Kievian melodies. It is in the Service of Burial which is due to be used at Windsor today.
Parts of it are used also in Greek Orthodoxy in the order for Nekrosimos. In my (Athens) Mikron Euchologion, the following rubric occurs before the text Meta ton hagion anapauson ... "It is the custom in the autocephalous Church of Greece that the following Kontakion be sung by the right hand choir."
I wonder what lies behind this way of putting things ...