Have a look at Church’s (first) Set Theory with a Universal Set, which is equiconsistent with ZFGC, though Church didn’t publish a full proof, and I think he abandoned the proof in his archives at Princeton. (My forthcoming doctoral thesis completes the proof for a variant with the singleton function as a set.) His archives also contain lecture notes on some more complicated theories with approximately the same name, but I believe he gave up on those theories, not just the details of the consistency proofs.
• Alonzo Church (1974). “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium. Proceedings of Symposia in Pure Mathematics XXV, ed. L. Henkin, American Mathematical Society, pp. 297–308.
• T. E. Forster (1995). Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford Logic Guides 31). Oxford University Press. ISBN 0-19-851477-8.
• T. E. Forster (2001). “Church’s Set Theory with a Universal Set.”
See also the Wikipedia article (disclaimer: I started it in its current form.)