After many more tries at Level 13 (which Beck assures me is the hardest of them all), I finally managed it. I didn’t realise previously, but different teams have different personalities, and the Blue Team were way more deadly than the Green Team. Killing the Blues off first, then, was the way to go.
After 13 was complete, reaching 17 was pretty simple. I only have one Achievement left to go now too – to complete Challenge Mode.