My pup was that small when I brought him home at 12 weeks. He could not hold it all night. He also pooped in the middle of the night. The first thing I did was change his food. I'm not sure this is true, but they say the higher the quality of food the less poopage they have.
I also took him out every night at 2am. I did this when I had him in the crate. I put him the crate at 10pm and took him out at 2am and again 6am. He now sleeps with me and has slept through the the night since. I drove myself crazy with wondering whether to take him outside or use the pads. I decided on outside because I think it was more natural to him. BUT I'm here all day at the house so that makes it easier. I put him in a gated area whenever we go out. He's still not completely housebroken. He had a pee accident yesterday and 2 the day before. My problem now is figuring out a way to get him to alert me to go outside.
Try feeding him on a set schedule then you may be able to time his poops.
I only kept in a crate at night for 2 wks (until he was 14 wks old). He started sleeping in my bed at 16wks and slept through the night. I think I made a mistake doing this because I can't get any sleep at night because he lays on my head.