It's not clear to me that a default judgement in a civil case against a moving company would be something their shipping insurer would pay out.

As is demonstrated in this blog post, the insurer paid out directly on a claim of damages, no civil court required. But above & beyond that, to recoup costs paid for the move based on a dispute about the service rendered, my understanding is that you'd end up with a judgement against the moving company and the same struggles chasing them down as you already had.