As expected you need to complete all four daily challenges each day for a week for a total of 28 daily challenges. Supposedly the count is supposed to reset on Sundays, but in my experience it's buggy and you may carry over progress. I currently have completed 38/28 daily challenges this week. This means that it may be possible to get it without doing the harder challenges or by starting in the middle of the week, if you're lucky.
Completing the impoppable challenge every day may be a challenge. My best tip would be to consider rerolling if you don't get any good damage dealing towers or a really hard map like tar pits. If you have started the challenge you can't change your mind and reroll afterwards.
The five round 85 ZOMGs are significantly easier to take down with heavy ability usage. Ground Zeros, Pirate Lords, Tech Terrors, MOAB annihilators, Spike Storms and First Strike Capabilities are all good damage dealers. If you need you can even sell and rebuy a tower to reset the cooldown. If all else fails buy bloonsday devices using monkey money which can if necessary solo the round. If you ever die you can press Alt+F4 on the death screen to quit out before your save game is deleted saving you from replaying the challenge.