I would recommend episode 2 for this achievement.
This is one of the hardest achievements in the game not because it is hard to win an episode without getting the time deduction, but because, even if you aren't trying to collect coins, chances are you are going to collect at least one between the qualifier, final four, and wipeout zone.
Important places where rings can be accidentally gotten: (if you fail pause and hit restart course, it won't void the achievement.)
There are silver rings on the second to last qualifier obstacle that you can easily accidentally get. If you do a full power jump and lean forward the whole time, you should miss all three.
There is a gold ring on the last qualifier obstacle that is even easier to get. It is tricky to get on the rope without hitting the coin. Go up to the edge of the platform and when the rope is swinging towards you, jump so you latch on. Now if you go all the way back you'll get the ring. This is sort of a blessing in disguise because if you swing with full power you'll hit the top of the hoop, so jump normally and you'll make it.
Let your A.I. friend go first on this one. Let them go ahead of you and let them collect all of the rings. If they miss a ring, do your best to avoid collecting it. If you collect it, you can restart.
The A.I. also needs to finish first because there is a ring at hone plate that is literally impossible to miss if you finish first.
The hardest part here is the first obstacle with the treadmills and bumpers. What I did was jump over the bumpers when they were extended (in your way) so there was no way you could collect the coin. Then I would put my hands out for half a second to prevent Mabel from accidentally running into the next coin.
The only other slightly hard part in the zone is the part where you get launched up by a trampoline and then are in a pit of big balls and mud. Just go to the very left to avoid getting the coin.
If you've made it this far you should be good with every other obstacle.
Good luck! And Big Balls.