This worked brilliantly on my data, thank you!! Of course, I hoped for a cleaner solution, but having an unlimited supply of random cov matrices definitely worth it. I did a quick comparison of the matrix traces between the original and generated covs 
Blue and orange are the data and random traces, respectively.
And this is the comparison of one of the off-diagonal elements:
The modes are reasonably close, the shapes are somewhat different, but of course the data is not random. It is still much closer than I would ever get by manual tinkering with distribution parameters. Now I just have to understand the role of the offset in nu, it is clearly necessary for NUTS to work. I would never guessed this, thank you for this insight.
Gergely