Oh, saying A,B,C,D are in [0,1] restricts quite a bit. It eliminates distributions with support over all the reals

???

There are easy to compute bijections from R to [0,1], etc.

The Bayesian approach to this problem uses an hierarchical distribution with two levels: one specifying the distribution p[A,B,C,D | X] in terms of some parameter vector X, and the other specifying the distribution p[X]

Yes, parametric Bayes does this. I am giving you a problem where you can't write down p(A,B,C,D | X) explicitly and then asking you to solve something frequentists are quite happy solving. Yes I am aware I can do a prior for this in the discrete case. I am sure a paper will come of it eventually.

Latent variables are still a pain, though.

The whole point of things like the beautiful distribution is you don't have to deal with latent variables. By the way the reason to think about H1 is that it represents all independences over A,B,C,D in this latent variable DAG:

A <- u1 -> B <- u2 -> C <- u3 -> D <- u4 -> A

where we marginalize out the ui variables.

which can indeed be written quite elegantly as an exponential family distribution with features for each clique in the graph

I think you might be confusing undirected and bidirected graph models. The former form linear exponential families and can be parameterized via cliques, the latter form curved exponential families, and can be parameterized via connected sets.