Although self-contained, this question is a follow-up to this earlier one. Also, thanks to Fedor Pakhomov for fixing a trivial early version of this question!

Work in $\mathsf{ZFC}$ + "There is a measurable cardinal," and let $\Omega$ be the least measurable. (Measurability is massive overkill here, I just want to make sure I have plenty of "room," so to speak.) Say that a sentence $\sigma$ is an inaccessibility safety net iff $V_\Omega$ satisfies each of the following statements:

• $\mathsf{ZFC+\sigma}$ has arbitrarily large transitive models, and

• whenever $M$ is a transitive model of $\mathsf{ZFC+\sigma}$ and $\alpha$ is $M$-inaccessible there is a transitive end extension $N\supseteq M$ such that $N\models\mathsf{ZFC}$, $\alpha$ is $N$-inaccessible, and $M\cap\mathsf{Ord}<N\cap\mathsf{Ord}$.

As far as I can tell, Farmer S's argument at the above-linked question - which relied on pointwise-definable models - does not prove that inaccessibility safety nets cannot exist (consider e.g. $\sigma\equiv\forall x[\mathsf{V\not=HOD}(x)]$).

Question: Is there an inaccessibility safety net?

In some sense an inaccessibility safety net has to axiomatize (a superset of) the consequences which the existence of a "large" inaccessible would have for a "small" inaccessible. Of course this is rather vague, but hopefully it helps motivate this question.