Of course, fully unrestricted unions in ZFC will immediately lead to inconsistency, but is there a variation where restrictions on allowed classes are given explicitly at least, in terms of the logical structure of the formula that defines the class? A particular class I have in mind is that of accessible cardinals, so Gödel-Bernays can't work since it is equivalent to ZFC.
In his original paper on ordinals Cantor asserts three "generation principles" for them. The third principle in rough translation is "if already generated ordinals share a property then they can be grouped together to form a new ordinal". He generates $\omega_1$ by using countability as the property on all ordinals generated after $\omega$. In ZFC this is justified by observing that countable ordinals can be a priori proved to form a set using the power set axiom.
There can't be such a justification for accessible cardinals of course, but the logical structure of countability doesn't seem to be much different from that of accessibility. Just postulating inaccessibles seems very ad hoc, from Cantor's original point of view they exist as much as $\aleph_1$.