Can I use Trait Doctoring to to make a Circle of Protection monocolor or multicolor?
No, it cannot.
As per the ruling on the card:

The type-changing effect can change part of a word such as “nonblack” or “swampwalk” if the part of the word is being used to refer to a color or basic land type.

Referring to color in general does not refer to a color.
As per the comp rules:

105.1. There are five colors in the Magic game: white, blue, black, red, and green.
105.4. If a player is asked to choose a color, he or she must choose one of the five colors. &quot;Multicolored&quot; is not a color. Neither is &quot;colorless.&quot;