This can be done in one game, not just one round.
I was able to get this in a few rounds by baiting out someone's snowball with my invulnerability. Once that was done, you follow them. Follow them for as long as possible. Try to be sneaky, but it's not always necessary. Most of the time they'll get bored of the chase and try to pick up snow. This is when you strike. Repeat this 3 times in one game and you'll have the achievement.
You can also hear people picking up snow, so keep an ear open for that too. I was able to pick off some people just by listening. I was able to achieve this in multiple games using this strategy, so hopefully this can help you get the achievement too!
Also, just like the other achievement, dying doesn't reset progress.
Let me know if there is anything I can answer for you or anything I can do to improve the guide.
Deschain suggests using your ult and firing a shot. Some people may not know what the ult does yet and may try to get some snow while yours is active.