ABC for bivariate betas

Crakel and Flegal just arXived a short paper running ABC for doing inference on the parameters of two families of bivariate betas. And I could not but read it thru. And wonder why ABC was that necessary to handle the model. The said bivariate betas are defined from

when

and

This makes each term in the pair Beta and the two components dependent. This construct was proposed by Arnold and Ng (2011). (The five-parameter version cancels the gammas for i=3,4,5.)

Since the pdf of the joint distribution is not available in closed form, Crakel and Flegal zoom on ABC-MCMC as the method of choice and discuss simulation experiments. (The choice of the tolerance ε as an absolute rather than relative value, ε=0.2,0.0.6,0.8, puzzles me, esp. since the distance between the summary statistics is not scaled.) I however wonder why other approaches are impossible. (Or why it is necessary to use this distribution to model correlated betas. Unless I am confused copulas were invented to this effect.) First, this is a latent variable model, so latent variables could be introduced inside an MCMC scheme. A wee bit costly but feasible. Second, several moments of those distributions are known so a empirical likelihood approach could be considered.