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 in 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 $\sf ZFC$.