Yes, in $L$ we put only constructible sets into $L_{\alpha+1}$, but you allow more sets to go into $\mathfrak{L}_{\alpha+1}$, which opens the door eventually to put any given set in, by first gradually adding its hereditary elements.

If you can articulate the desired features of the operator that make it shift-like, then the problem will become: how to prove there is no such operator.

You can define $\Box a$ to move everybody down to a place where it would be a violation of direct comparision, if possible. This works, but it is a bit like the ad hoc solution in my first comment, so perhaps not what you want.

Shifting within each $\omega$-block does not work, since this will be equivalent to being ascending within each block, but not necessarily overall.

Your question admits unwanted answers, where $\Box a=a$ if $a$ is ascending, otherwise something not above $a$. So presumably you want to say something more about the nature of the operator to be acceptable.

My two axioms have that feature. And yet jointly they imply V=L.

You ask two questions, but a negative answer to the first amounts to a positive answer to the second, which makes yes/no confusing.