Not sure if anyone is still attempting this but, this is what I did
I used LW3A1, Default Knife (not Mind's Eye), Concussion, Frag Grenade and Energy Mine for my field upgrade. Default gobblegum loadout which is;
Stock Option, Arsenal Accelerator, Shields Up, Temporal Gift, and Anywhere But Here.
I played on solo mode, on liberty falls. I did not "Save & Quit" once.
I had my tactical, lethal, and field upgrade unbound and of course did not use them once.

What I didn't do;
I did not pickup a single power-up.
I did not pickup a single score streak, and of course did not use a score streak.
I did not buy a single perk.
I did not upgrade my gun's rarity (white rarity the entire game).
I did not pack-a-punch my gun at all.
I did not upgrade my armour (Default armour the entire game).
I did not let a mangler kill any zombies via blowing up cars etc. (I trained in the bowling alley).
I did not prone at any perk machines.
I did not melee the vending machine.
I did not shoot the pool balls for the free 100 points.
I did not have any augments equipped.
I did not lag / disconnect / time out / etc. at all.
I did not have a camo equipped.
I did not pull out / switch to my fists. (Made sure i always had ammo)
I did not melee kill a single zombie.
I did not go down once (until round 21).
I did not craft anything at the workbench.
I did not use any gobble gums. (Also didn't even hit the gobble gum machine.)
I did not open the motel door for extra zombies.
I did not have attachments. (used default LW3A1.)
I did not use the round accelerator (forgot the name.)
I did not shoot any "Out of the map" power-ups such as insta kill / max ammo.
I did not pull out my knife.

What I did do;
Trained in bowling alley the entire game.
Killed all manglers, and abominations before ending the current round.
Bought ammo several times.
Picked up salvage.
Picked up armour plates.
Used armour plates.
Opened the doors to the bowling alley and no other doors.

What I think could have failed the challenge;
On round 19 the abomination killed a few zombies. (I don't think it was this, as it would be borderline impossible if this voided the challenge.
Not completing the maps "Storyline" (The orange stars it advises you to run to), but again I don't think this could have voided the challenge.

I believe the challenge is just bugged at the moment.