My puppy was litter trained. I had an xpen set up in my bedroom with her crate and a litter box for when I was at work. Worked like a charm. She never went to the bathroom in her crate and when good weather came I had no problem transitioning her to completely using the outdoors. Like Tayda's mom, I too used Equine Pine (you can get it cheap in big bags at Tractor Supply - much cheaper than the similar Feline Pine).
With the doggie door and your b going out on its own to use the litter box, I would say that might be a stretch - especially if it is a puppy. I know my pup was never trusted to go on her own without me outside watching her. She would have been too distracted in the yard (she's a busy girl :D) and if I didn't watch her I know she would have come back in and pee'd. Many threads on here of new pup owners that have that happen, so instead of leaving it to chance, I just made sure to see her go to the bathroom.