I would definitely add at least 1 more litter box where the puppy has no access, it's possible the cats got spooked, either by the puppy or something else, while going through the new cat door.
If the cats must pass the puppy's kennel to go to the current litter box it might be a good idea to move the kennel farther away from that door to see if that helps.
What are you cleaning the pooped on areas with? A good enzymatic cleaner is needed to remove all scent that we can't smell but they can. Nature's Miracle is one or you can make your own with: 1 cup water, 1/4 cup white vinegar, 1/4 cup isopropyl (rubbing) alcohol, mix together in a clean/new spray bottle. Spray the area affected, blot with paper towel/clean rag, spray again & let dry.
How does the puppy react to the cats? Especially if they are scared & run?
Is the puppy leashed inside the house for quick calm corrections if he gets too close for the cats comfort level or attempts to chase?
Are you teaching the puppy a "leave it" or similar command around the cats?
Avoid biting when a simple growl will do
The Spirit Lives As Long As Someone Who Lives Remembers You - Navaho Saying
Vindication ~ For all those pets who became sick or lost their lives from tainted pet food