Hi. I'm new here, but have had cats in my life for a long time. Just a couple of thoughts. These have all happened to us. Please don't take offense.
Hope you can solve the mystery. You sound like you really love your cats or you wouldn't be here asking for help. Good luck!
1. Is the litter box big enough? Don't know the size of your cat, but I have had big cats before that thought they were pooping in the box, when their rear ends were actually hanging outside of the box. Solution: Bigger litter boxes.
2. Are you sure cat #1 isn't forcing cat #2 out of the litter box or harrassing cat #2 so he can't poop in peace, but is actually rushed? Cats do have a pecking order.
3. What are you using to clean the litter box? Are you scrupulous about cleaning feces off the sides of the box? Does the litter box smell of urine? Cats have a fantastic sense of smell and can detect odors we cannot? My cats won't use the litter boxes when they are due for changing and start to smell of urine. I use a nontoxic cleaner, Sun & Earth.
4. Scooping or clay litter? Since I have multiple cats and use newspaper pellet litter, I completely change the litter twice a week and scoop stool on a daily basis. Two of the boxes that get the most use have to be changed twice a week.
5. Sometimes if one cat is sick (you said one cat had a problems previously), the other cat(s) won't use the box because there is an off smell.
6. Has both cats been checked by a vet for intestinal or urinary problems?