Okay....you say at 3.20 am your sister takes the dogs outside....yet she has to be at work at 3.30! I don't obviously know how long it takes her to get to work. It might only be two minutes. But that is an average of five to eight minutes outside. Or less??
Has your dog had enough time to sniff about trot around, go round in circles (like they do) and do his business? They can't always poop to order any more than we can. If someone said to you "You have five minutes to poop right now." Would that work for you?
What sort of exercise and proper structured walks does your dog get during the day, and does he get a walk a few hours after his evening meal....say 20-30 minutes?
If you live in an area where night walking is unsafe, then the best plan is to walk him in the day and feed him his main meal early afternoon, to give his digestion time to work. But make sure he gets proper walks. Like us -ofetn a little exercise can make them "go".
I am looking firstly at any obvious physical reasons rather than behavioural.
If the other dog doesn't do the same, then again -it's like us! All dogs are individuals and have different ways and different bodily needs.