I doubt it.
A potential lawsuit would have to be placed against the federal government. All levels of government, local, state and federal enjoy strong sovereign immunity from lawsuits.
At the federal level, that immunity is probably absolute. Thus, a meaningful lawsuit maybe impossible to place.
Governments can agree to enter into settlements for a variety of motivations, including fairness, avoidance of bad publicity, cost savings, and political pressure. Actually forcing them to pay a settlement, however, is extremely difficult and at the federal level, it could well be impossible.
I am also generally aware that it would be extremely rare, or perhaps never happening for the DOD and the various military branches to agree to pay settlements beyond the military death benefit as they would be subject to endless lawsuits for every training death and every battle death.
In the end, my guess is that the military is going to offer the active duty death benefit, then the matter will be financially closed as far as they are concerned.