We say that a model $M$ of $\mathsf{ZF}$ satisfies Small Violations of Choice ($\mathsf{SVC}$) if all (any) of the following apply:

1. There is a model $V\subseteq M$ such that $V\vDash\mathsf{ZFC}$, and $M$ is a symmetric extension of $V$.
2. There is a forcing $\mathbb{P}\in M$ such that $\mathbb{P}\mathrel{\Vdash}\mathsf{AC}$.
3. There is $A\in M$ such that for all $X\in M$, there is an ordinal $\eta$ and a surjection $f\colon A\times\eta\to X$ in $M$.
4. There is $A\in M$ such that for all $X\in M$, there is an ordinal $\eta$ and an injection $f\colon X\to A\times\eta$ in $M$.

I am particularly interested in point 2, and in forcing $\mathsf{AC}$ in such a way that if $\lambda\neq\kappa\in M$ are well-ordered cardinals, $\mathbb{P}\mathrel{\Vdash}\check{\lambda}\neq\check{\kappa}$.

Is this always possible?