I have been looking for a source for this concept, as the Minhag that I’ve seen in Shuls is that they do say Hakohen even in the Mi Shebeirach for the ill. Would the rabbi please guide me to the source of this in the books of Halacha or Minhag?

Shalom, I too have not seen this exact detail addressed explicitly in a halachic book. My answer is based on the fact that in many sources - and siddurim - the language used for an aliyah to the Torah is "ploni ben ploni HaCohen", whereas the language used for the misheberach is just "ploni ben ploni". Look in your siddur and you will see that for the aliyah it prints the word "HaCohen", but not in the misheberach. See for example the Mishna Brura 116, 3, who writes to pray for a sick person as just "ploni ben ploni", as opposed to the language of the siddurim of the Rishonim and the Machzor Vitri which all say "ploni ben ploni HaCohen" for the aliyah to the Torah. This makes logical sense. The name used for the aliyah is to publicly honor the person, and so we use all the titles of respect and lineage. But the misheberach prayer is just that - a prayer. When praying we use the name to satisfy the injunction to mention the sick person’s name when praying for them (if they are not in front of us at the time). Here titles serve no purpose. Blessings, D. Sperling.