From what I understand, the POT library solves 4.1 (Entropic regularization of the Wasserstein distance, say W(p,q) ), deriving the gradient in 4.2 and the relaxation in 4.3 (first going to W(p_approx,q_approx)+DKL(p_approx,p)+DKL(q_approx,q) and then generalising DKL to allow p/q approx to not be distributions seems to go beyond that.
Funny that they use the difference between MNIST class labels as a metric for the target. That reminded me of your regression approach.