How badly does foundation fail in NF(etc.)?

This is quite messy, actually. NF doesn't prove the existence of many transitive sets, and the restriction of $\in$ to a transitive set is a set very rarely. So your set TRAN might turn out to have very little in it. It might not contain any infinite sets for example. Specifically the restriction of $\in$ to the transitive set $V$ is provably not a set, so the graph of which $V$ is a picture literally doesn't exist. I suspect the question you have at the back of your mind is subtly different.

If you are interested in how badly foundation fails in NF(U), try looking at it like this. We know that $V \in V$, so foundation fails. That is: there are classes (sets, even) that lack $\in$-minimal members. One might ask: is $V$ the only reason for the existence of such collections? Might it be the case that every (``bottomless'') set lacking an $\in$-minimal member contains $V$?. Such a hypothesis would exclude Quine atoms, and we know how to exclude them anyway. At this stage it seems possible that we could find models of NF in which every bottomless class contains $V$ - so that the existence of $V$ is the only thing that falsifies Foundation.

Is this the kind of thing you are after?