Your mother definitely has no standing, because she is the great-aunt, not the grandma, if I follow your story correctly.
I'm sorry that your mom is upset, and understand that she may not think the move is in the child's best interests, but I fail to see how anything would be improved if she were to drag the non-support-paying exbf into it, in an attempt to what? Sue for custody? Get ahold of C and give him back to his great-aunt to raise? Not likely.
Your cousin may be a total jerk to manipulate your mom like this, but unless there is an existing custody order, she has the right to move with her child wherever she wishes. You are also have the right, as a member of the human race, to tell K that she's planning to leave - but I don't know that you would get the result you're hoping for. Are you even positive he's the biofather? Is he on the birth certificate?