$\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication A. S. Daghighi, M. Golshani, J. D. Hamkins, and E. Jeřábek proved in "The foundation axiom and elementary self-embeddings of the universe" that, working in ZFGC$^{\text{−f}}$+BAFA, there are nontrivial automorphisms and elementary embeddings of the universe $V$ into itself.
Accordingly, Kunen inconsistency is circumscribed for this class of ill-founded theories.
Does it follow that $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ is a non inconsistent extension of $\text{ZFGC}^{\text{−f}}+\text{BAFA}$?
If so, is it known which of the large cardinal properties would $\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ imply?
 A: I'm glad to hear you're reading our paper, which can be found here: The foundation axiom and elementary self-embeddings of the universe. Click through to the arxiv for a pdf — and I note that the title you mention is from an earlier draft of this article, so you may want to look at the updated version of the article.
In theorem 1 of the article, we prove in $\text{ZFC}^{-f}$ that any $\Sigma_1$-elementary embedding $j:V\to V$ must fix every well-founded set, and in particular, every ordinal. This is the residue of the Kunen inconsistency in this foundation-ness context. It follows, however, that although as you mention BAFA proves the existence of nontrivial elementary embeddings $j:V\to V$, we do not get nontriviality on the ordinals, and so there are no Reinhardt cardinals here to be found. The embeddings provided by BAFA have no critical points.
Further, since the consistency strength of BAFA is no greater than that of ZFC, it follows also that consistency-wise, one cannot provably get any large cardinals from such an embbedding in $\text{ZFC}^{-f}$. 
