Short and simple, the bug is that you receive NO mission progress at all from anything that you did in the first round of a Stopwatch match. I had noticed weird things happening with mission progress before when I played Stopwatch, but I didn't really try to crack down on the cause until today. To prove it I did a simple test with my current combat xp mission: Go ham the first round, then do no harm the second.
As you can see in these screenshots, I killed 28 and topped my team in the first round. Then, the next round, I went Aura and did nothing but heal and revive - I never engaged in combat. We lost the round, and then in the after-game report my fears were confirmed - Not one of my 28 kills in the first round was counting towards the Hunter Killer mission. And it's not that my missions are totally borked either, since you can see me get progress for Operation: Proxy.
PLS FIX STOPWATCH SPLASH DAMAGE