What are the Halachic basis for this ruling? What are the mekorot for this halacha?

Shalom, Thank you for your question. The words “HaCohen” or “HaLevi” are not part of a person's name, rather they are a title, similar to “HaRav”. When praying for the sick we use their name without title. The source for not using titles when praying for the sick can be found in Sefer Chasidim (chapter 800), where he writes “One who is praying for his sick father should not say "Heal my father my master" or "To my master my father heal". That is to say, one does not use the usual honorific titles usually added to a name when praying for the sick. (The Sefer Chasidim is referring to the titles added when using their father's name (Avi Mori – My father, my teacher) – but the same would hold true for all titles – such as The Cohen or Levi). He goes on to write that the same is true for the titles one usually adds for their teacher or Rabbi – these too should be left out when praying for the sick. The Birkei Yosef (Yoreh De'ah 242) explains the Sefer Chasidim - that there is no place to mention special stature when praying to Hashem. This would explain that the regular titles we add when calling someone to the Torah reading (Rav, Cohen, Levi etc), are to honour the person in the eyes of the community – but when praying for them towards Hashem, the concept of honour does not apply. All are equal in the eyes of Hashem. Blessings.