The main Conjecture (MC) has at least 3 different equivalent formulations. To ease the notations in your question, since here $p\nmid [\mathbf Q(\mu_p):\mathbf Q]$, we may as well sum up the isotypical components relative to even (odd) characters and deal only with the $\pm1$ components of the modules involved.
(MC$1$) Sticking to your notations, the MC reads char(${C_{\infty}}^+)$ = char (${E_{\infty}}^+/{V_{\infty}}^+)$, and you ask whether this could be extended to the minus part. No, because the minus parts of the units at finite levels are reduced to $\pm 1$, so upstairs char(${C_{\infty}}^-)$ would be trivial, whereas downstairs the arithmetic of the minus part of the $p$-class group is definitely not (Kummer, Bernoulli numbers, etc.) On the opposite, the plus part is conjectured to be trivial : downstairs, it is Vandiver's conjecture that $p\nmid h^+$; upstairs, Greenberg's conjecture that ${C_{\infty}}^+$ is finite. To have an idea of the strength of these conjectures, note that Greenberg's (the weakest one) easily implies directly the MC. Actually (MC$1$), which avoids the appeal to $p$-adic $L$-functions, is rather specific of the proofs of the MC based on Euler systems.
(MC$2$) In view of the comments above concerning (MC$1$), it appears that - at the time being - our true knowledge of the arithmetic of the $p$-class groups is rather one-handed. To stress the most striking feature, the link with $p$-adic $L$-functions, we must come back to characters, since the trivial character must be excluded because of the pole at $s=1$. The character-wise formulation (MC$2$) on the minus size then reads : Let $\omega$ denote the Teichmüller character. For any non trivial even character $\chi$ of Gal($\mathbf Q(\mu_p)/\mathbf Q)$, the characteristic series $f_{\chi}$ of the $\omega \chi^{-1}$-part of ${C_{\infty}}$ satisfies $f_{\chi}((1+p)^s -1)=L_p(\chi, s)$ for all $s\in \mathbf Z_p$ .
(MC$3$) To come back to the plus size we must change modules, introducing ${B_{\infty}}$= the Galois group over $\mathbf Q(\mu_{p^\infty})$ of the maximal pro-p-abelian extension of $\mathbf Q(\mu_{p^\infty})$ which is $p$-ramified, i.e. unramified outside $p$. Then (MC$3$) reads : For any non trivial even character $\chi$ of Gal($\mathbf Q(\mu_p)/\mathbf Q)$, the characteristic series $g_{\chi}$ of the $ \chi$-part of ${B_{\infty}}$ satisfies $f_{\chi}((1+p)^{1-s} -1)=L_p(\chi, s)$ for all $s\in \mathbf Z_p$ . Comparing with (MC$2$), note the shift between the pair ($s,\omega \chi^{-1}$) and the pair ($1-s, \chi^{-1}$), which is due to a combination of class-field theoretic iso. and Kummer duality, usually called Spiegelung ./.
Best Answer
I keep all your notations but I'll introduce also the $\Lambda$-modules $X$ (rep. $X'$) = Galois group over $K$ of the maximal abelian pro-$p$-extension of $K$ which is uramified (resp. is unramified and splits totally all the $p$-places of $K$). It is known that the Pontryagin dual of $C(p)$ is pseudo-isomorphic to $X^*$ = the module $X$ with inverted action of $\Gamma = \bar{<\gamma>}$. The same property holds for $X'$. It follows that the finiteness of $C(p)^{\Gamma}$ is equivalent to that of $X^*_{\Gamma}$, i.e. $T^*=\gamma^{-1} -1$ does not divide the characteristic series of $X^*$, or equivalently, ${X^*}^{\Gamma}$ is finite, or equivalently, ${X}^{\Gamma}$ is finite. So we are brought back to construct an example of an infinite ${X}^{\Gamma}$.
Suppose moreover that $k$ is CM and $p$ is odd. Let $r$ be the number of $p$-places of $k^+$ which split completely in $k$. It is known that the kernel of the canonical surjection $X^- \to (X')^-$ is isomorphic to a direct sum $r$ factors of the form $\Lambda/\omega_{n_i}\Lambda$, where $\omega_n = {\gamma}^{p^n}-1$ (a more precise statement is in the appendix of [FG]). This implies that $(X^-)^{\Gamma}$ is infinite, and we are done.
[FG] L.-J. Federer, B.-H. Gross: Regulators and Iwasawa modules, Invent. Math, 62 (1981), 443-457