Analogue of $k$-wise independence for other distributions than uniform
I met once a similar notion to what you want in one paper by Mark Zhandry's paper under the name $k$-wise equivalent. I cannot find further references or pointers from that paper, but I think this name nicely describes your something.
Concretely, the original definition in the paper is about the distributions over the functions $f: X\rightarrow Y$, and define the size-$k$ marginal by its marginal distribution for any size-$k$ subset $W$ of $X$. The notion $k$-wise equivalence is defined accordingly, that is, two distributions are identical on any $k$-marginal.
It is not directly addressing your main question, but we can say that your desired distribution is $k$-wise equivalent to the uniform random permutation. It also, unfortunately, does not give any efficient construction for equivalent distribution as well.