@Suvrit, am curious as to why you say ($*$) is not a reverse Minkowski inequality (does such a thing have a precise definition?). That ($*$) follows from Minkowski (plus Fiedler's, plus linearizing ($**$)) should not preclude such a thing. E.g., 'reverse Holder' follows from Holder's inequality. In fact, ($*$) actually implies Minkowski's inequality, so could be regarded as an improvement. Indeed, Minkowski is equivalent to concavity of $A\mapsto (\det A)^{1/n}$, or $f(t)\leq f(0) + t f'(0)$ for $t$ small, and $f(t) := (\det(A+t I))^{1/n}$. But, this is precisely ($*$) with $B=tI$.

I think it works by just considering positive square roots: $C = C^{1/2}C^{1/2}$. Then, the argument goes through as Suvrit suggested by considering $\tilde{A} = C^{-1/2}AC^{-1/2}$, $\tilde{B} = C^{-1/2}BC^{-1/2}$, and $\tilde{C}=I$.