Here is the question:
Are wounds assigned as a trigger of losing the skirmish? or are the assigned as a step (ie 1) who won, 2) assign wounds)?
If they are assigned as a trigger (they'd be manditory), the the FP decides which happens first: the wounds from losing or the trigger from stooping to kill.
If it is stepwise, then stooping to kills trigger hapens before assigning the wounds.
I don't think the rules are clear on this. I'd prefer that wounds (or overwhelm) are triggeres of losing, but that's not rule at all.