Paulo (from the Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil) has posted an answer to my earlier question both as a comment on the ‘Og and as a solution on StackOverflow (with a much more readable LaTeX output). His solution is based on the observation that the multidimensional log-normal distribution still allows for closed form expressions of both the mean and the variance and that those expressions can further be inverted to impose the pair (μ,Σ)  on the log-normal vector. In addition, he shows that the only constraint on the covariance matrix is that the covariance σ_ij is larger than -μ_iμ_j.. Very neat!

In the meanwhile, I corrected my earlier R code on the gamma model, thanks to David Epstein pointing a mistake in the resolution of the moment equation and I added the constraint on the covariance, already noticed by David in his question. Here is the full code:

sol=function(mu,sigma){
  solub=TRUE
  alpha=rep(0,3)
  beta=rep(0,2)
  beta[1]=mu[1]/sigma[1]
  alpha[1]=mu[1]*beta[1]
  coef=mu[2]*sigma[1]-mu[1]*sigma[3]
  if (coef<0){
   solub=FALSE}else{
    beta[2]=coef/(sigma[1]*sigma[2]-sigma[3]^2)
    alpha[2]=sigma[3]*beta[2]/sigma[1]^2
    alpha[3]=mu[2]*beta[2]-mu[1]*alpha[2]
    if (alpha[3]    }
list(solub=solub,alpha=alpha,beta=beta)
}

mu=runif(2,0,10);sig=c(mu[1]^2/runif(1),mu[2]^2/runif(1));sol(mu,c(sig,runif(1,max(-sqrt(prod(sig)),
-mu[1]*mu[2]),sqrt(prod(sig)))))

and did not get any FALSE outcome when running this code several times.