Here is a model of your theory. Start with a countably infinite set of objects $X=\bigsqcup_n X_n$, partitioned into infinitely many infinite sections. We inductively define $\in_n$. Consider the various formulas $\phi^n(y,z)$ that use only at most $\in_m$ for $m<n$, with parameters $z$. With the various possible parameters $z$, these define certain subsets of $X$, but only countably many. For each such definable subset, which has not already arisen earlier, assign an object $x$ from $X_n$ to represent it, and define $y\in_n x\iff\phi^n(y,z)$ for this instance. That is, I pick the object to represent the set for each definable set, not for each formula. Thus, we will satisfy comprehension. The objects in $X_n$ will only have $\in_n$ members, and so no set has elements at more than one level, and so in this way we satisfy membership. Extensionality holds since we used one object for each definable set, using it at the earliest stage it becomes definable, and the object had elements only at that level.
This method of argument is rather similar to some of the solutions of Frege's Basic Law V, as discussed in my paper
- Fregean abstraction in Zermelo-Fraenkel set theory: a deflationary account, Annals of Mathematics and Philosophy, 2023. (arxiv:2209.07845)
See the prior art section there, where I discuss this method, used in various work by Parsons, Bell, and Burgess.
One can perform the construction over any given model already. That is, we can assume $X$ already has some other first-order structure, but then chop it into the sections and undertake this construction on top of it, allowing formulas $\phi^n$ in the full language. This is essentially how the Parsons method works.