Work within $\sf NBG$, define *ultraset* as:

$ultraset(x) \iff x \in V \lor x=V$

Define a new membership relation $\in^*$ over ultrasets as:

$y \in^* x \iff ultraset(y) \land (x \neq V \to y \in x) $

Now it can be proven in $\sf NBG$ that the world of ultrasets would satisfy the existence of a universal set, that it: $$ \exists x \forall y (y \in^* x)$$
And that all axioms of ZF with all of its quantifiers bounded by non universal sets [i.e. written as $\forall x (\neg \forall y (y \in^* x) \to...); \exists x (\neg \forall y (y \in^* x) \land ...)$], and each symbol $\in$ replaced by $\in^*$, would hold over the world of ultrasets!

Now take the axioms of your theory to be the universal set axiom and the re-written axioms of $\sf ZFC$ as above, and I think this would be equi-consistent with $\sf ZFC$, since $\sf NBG$ itself is conservative over $\sf ZFC$.

There are other tricky ways of doing that also. For example also work in $\sf NBG$, define a function $j$ that sends the empty set to $V$, sends each nonempty Zermelo natural to its predecessor, and fix all sets otherwise. 

Now define a membership relation $\in'$, over domain $V$, as: $$y \in' x \iff y \in j(x)$$, so the $\in$ empty set would turn into the universal $\in'$ set.

We can define a predicate "$Original$" to be the subclass of $V$ that is the hierarchy raised over $1$, i.e. its starts with set $1=\{0\}$  and each successive stage replaces $\{0\}$ instead of $\emptyset$, that is we have $V^*_{\alpha+1}= (\mathcal P(V^*_\alpha) \setminus \{\emptyset\}) \cup \{1\}$. 

Now we return to our world which has $V$ as its domain and $\in'$ membership relation, now clearly the $Original$ sets all belong to that world, and it is straighforward to see that $\sf ZFC$ axioms would hold over the world $Original$ with respect to $\in'$, i.e. if we re-write all axioms of $\sf ZFC$ restricted to $Original$ sets, and with $\in'$ replacing $\in$, then all of those would hold. 

So, this theory would also interpet $\sf ZFC$ over the $Original$ sector of its world. And I also think it would be equi-consistent with $\sf ZFC$ since it is in whole definable in $\sf NBG$ which is a conservative extension over $\sf ZFC$