The property you talk about the universe having is an interesting one, but I don't think causality is the right word for it. You've smuggled an extra component into the definition: each node having small fan-in (for some definition of "small"). Call this "locality". Lack of locality makes causal reasoning harder (sometimes astronomically harder) in some cases, but it does not break causal inference algorithms; it only makes them slower.

The time-turner implementation where you enumerate all possible universes, and select one that passed the self-consistency test, can be represented by a DAG; it's causal. It's just that the moment where the time-traveler lands depends on the whole space of later universes. That doesn't make the graph cyclic; it's just a large fanin. If the underlying physics is discrete and the range of time-turners is time limited to six hours, it's not even infinite fanin. And if you blur out irrelevant details, like we usually do when reasoning about physical processes, you can even construct manageable causal graphs of events involving time-turner usage, and use them to predict experimental outcomes!

You can imagine universes which violate the small-fanin criterion in other ways. For example, imagine a Conway's life-like game on an infinite plane, with a special tile type that copies a randomly-selected other cell in each timestep, with each cell having a probability of being selected that falls off with distance. Such cells would also have infinite fan-in, but there would still be a DAG representing the causal structure of that universe. It used to be believed that gravity behaved this way.