EDIT . For $n=2$, the smallest $C$ is as follows. All calculations use the library Groebner of Maple.
Let $a\approx 0.3787$ be the number satisfying $4a^8+27a^6+34a^4-117a^2+16=0$. Note that $a$ can be calculated by radicals. Then
$min(C)=\dfrac{\sqrt{2a^2+4+2a\sqrt{a^2+4}}+\sqrt{2a^4+6a^2+4+2a(a^2+1)\sqrt{a^2+4}}}{2(a^2+1)}$.
Moreover $min(C)\approx 2.184596$ is a root of the following polynomial: $20736x^8-122832x^6+126913x^4-62224x^2+1024$. Note that $min(C)$ can be calculated by radicals.