Hm, I’m loath to work with so many correlation parameters. I have a different train of thought: let’s stick with that idea of defining an auxiliary parameter \theta=CV_{total}^2+1, so we can define a prior for CV_{total} and work with \theta to derive the individual CV_i. Let’s specify a Dirichlet prior on a simplex parameter \kappa_{1:3} of length 3 (way fewer parameters: 3 instead of 8), and then let:
CV_{total}=\sqrt{(CV_1^2+1)(CV_2^2+1)(CV_3^2+1)-1}
CV_{total}^2+1=(CV^2_1+1)(CV^2_2+1)(CV^2_3+1) = \theta
Now let: \theta = \Pi_i \theta^{\kappa_i}, where \theta^{\kappa_i} = CV_i^2+1
Because \Sigma_i \kappa_i = 1, it follows that \Pi_i \theta^{\kappa_i} = \theta^{\Sigma_i \kappa_i}= \theta^1, which is \theta. And because \kappa_i \geq 0 and \theta \geq 1 by definition, it follows that CV_i^2+1 \geq 1, which is exactly the condition we need to be able to safely derive CV_i=\sqrt{\theta^{\kappa_i}-1}=\sqrt{(CV_{total}^2+1)^{\kappa_i}-1}.
I think this may be it!! This is a way (maybe the most straightforward?!) to soft-constrain the product of a bunch of parameters. I’d rather work with those few hyperparameters of the Dirichlet distribution to get the desired pushforward distribution for each CV_i than dealing with N(N-1)/2 correlation parameters to tweak a pushforward distribution for the product of N parameters. Am I missing something?