(This answers what was the original question, which was with $\mathrm{Coll}(\omega,{<\kappa})$ replacing $\mathrm{Coll}(\omega,\kappa)$.)

Hmm...doesn't this contradict Theorem 1.22 of "MICE WITH FINITELY MANY WOODIN CARDINALS
FROM OPTIMAL DETERMINACY HYPOTHESES" by Mueller, Schindler, Woodin? According to that result, under ZFC, $M_1^\#$ exists and is $\omega_1$-iterable iff boldface-$\Pi^1_1$ and lightface-$\Delta^1_2$ determinacy holds. (So work in $V=M_1$, and let $\delta$ be the unique Woodin cardinal. Let $G$ be $(V,\mathbb{P})$-generic where $\mathbb{P}=\mathrm{Coll}(\omega,{<\delta})$. Suppose lightface-$\Sigma^1_2$ determinacy holds in $V[G]$. Then lightface-$\Delta^1_2$ determinacy holds there. But also, every real has a sharp in $V[G]$, so boldface-$\Pi^1_1$ determinacy holds there also. So $V[G]\models$"$M_1^\#$ exists and is $\omega_1$-iterable". But then $(M_1^\#)^{V[G]}\in\mathrm{HOD}^{V[G]}$, so by homogeneity of the collapse, $(M_1^\#)^{V[G]}\in V=M_1$. But then $M_1\models$"$M_1^\#$ exists and is $\delta$-iterable", which is impossible.)