Set theories combining ZFC and NFU

Question

I know very little set theory, but I have heard of NFU before and am wondering whether it can be used as a "platform" to "host" an arbitrary first-order theory in its urelements, in this case ZFC.

I sometimes talk about class-sized things (and frequently make mistakes when doing so), so I am in the market for a set theory that supports multiple levels of classes or something similar like the flattened hierarchy of NFU, even if the resulting classes are less flexible than sets.

I'm wondering whether this strawman theory is consistent given that ZFC and NFU are both consistent individually. I'm also curious whether there's a real way to combine these theories together so you have a set theory with ZFC at the bottom rung and more rigid classes on the higher rungs.

The alternative set theory NFU can be axiomatized with two axioms and an axiom schema. An axiom of infinity or axiom of choice can also be added without breaking consistency as far as I know.

1. $\varnothing$ does not contain any elements.
2. Any entities that contain at least one element are equal if their elements are equal.
3. Let $\varphi$ be a stratified formula with parameters. There exists a set $X$ such that all entities $e$ are in $X$ if and only if they satisfy $\varphi$.

Superficially, sets in NFU seem very classlike to me. You can have a set of all groups and even a universal set as long as you respect the quasi-type-system when invoking comprehension.

This made me wonder whether we can combine NFU and ZFC into a Frankenstein theory where ZFC holds for the urelements.

To that end, here is the axiomatization for NFU+ZFC. I introduce two binary relation symbols $\in$ and $E$. ZFC gets $\in$ and NFU gets $E$. In prose, $xEy$ is written $x$ is an $E$-element of $y$ or $y$ $E$-contains $x$.

I'll define a set as an entity that does not $E$-contain anything.

An entity is defined to be a class if and only if it does not $\in$-contain anything.

1. An entity is a set and a class if and only if it is $\varnothing$.
2. Every entity is a set or a class.
3. If a set contains a class $k$, then $k$ is $\varnothing$.
4. Any entities that $E$-contain at least one element are equal if their $E$-elements are equal.
5. Let $\varphi$ be a stratified formula with parameters, unlike $E$, both arguments to $\in$ are constrained to have the same type. There exists a class consisting of all entities satisfying $\varphi$.
6. All the axioms of ZFC with quantifiers replaced in such a way to constrain their bound variables to be sets and all parameters constrained to be sets as well.
7. $\varnothing$ is the empty set in ZFC.

So, the both-set-and-class $\varnothing$ is a bit strange, and needing to juggle two elementhood relations is inconvenient, but I'm curious whether this approach of stapling two set theories together is valid in general or has any hidden drawbacks.