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 some recursive wellorder $W$ of $\omega$ in ordertype $\alpha$ with $x\in L_\alpha$; then $T^+$ models ``$W$ is a wellorder'', and can recover $x$ from $W$.) But with $T^+\in L_{\omega_1^{\mathrm{ck}}}$, this gives a contradiction.