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Claim: $T+$"$T$ is $\omega$-complete" is inconsistent. For suppose it's consistent and now work in a model $V$ of this theory. Let $T^+$ be the resulting completion of $T$ (i.e. the unique theory of the $\omega$-models of $T$ in the sense of $V$). Then note that $T^+$ is a $\Delta^1_1$ real, so $T^+\in L_{\omega_1^{\mathrm{ck}}}$. But $L_{\omega_1^{\mathrm{ck}}}\subseteq\mathrm{wfp}(M)$ whenever $M\models T$ is an $\omega$-model, and therefore every real $x\in L_{\omega_1^{\mathrm{ck}}}$ is such that $x\leq_{\mathrm{T}} T^+$ (the sub-$\mathrm{T}$ there being "Turing", as opposed to the theory $T$). (Given $x$, fix a wellorder $W$ of $\omega$ in ordertype $\alpha$ with $x\in L_\alpha$. Then (roughly) $T^+$ models "$W$ is a wellorder", and can recover $x$ from $W$. Formally, fix an integer $e$ which indexes a recursive wellorder of $\omega$ in ordertype $\alpha$ with $x\in L_\alpha$. Recall $L_\alpha$ projects to $\omega$, and $x\leq_{\mathrm{T}} t^{L_\alpha}$, the first-order theory of $L_\alpha$. Fix a Turing reduction $n$ of $x$ from $t^{L_\alpha}$. Then for $m<\omega$, we have $m\in x$ iff $T^+$ contains the statement "Let $\beta$ be the ordertype of the wellorder coded by $e$, and let $y\leq_{\mathrm{T}} t^{L_\beta}$ via the $n$th Turing program; then $m\in y$".) But with $T^+\in L_{\omega_1^{\mathrm{ck}}}$, this gives a contradiction.