How to deal with a puppy who will not poop outside?

She's four months old; we've had her for two. The first several weeks were that unbearable cold/storm/unbearable cold/ice storm/unbearable cold thing a bunch of you probably dealt with as well, and not great for either humans or small animals to be outside for long periods, so we concentrated on puppy pads. She does great with those.

After about a month, when the weather started being tolerable, I took her outside to begin housetraining. I followed all the recommendations: right after waking, after eating, and then about every hour while she was awake. Fifteen minutes at least each trip. Timed meals whenever possible. Contained in a pen or in her crate for most of the rest of the day. Supervised playtime with us keeping an eye on her for pee & poop activity. For a few days we had great success and there were a couple of outdoor poops each day and lots of outdoor pees.

But now she outright refuses to poop outside. At first she refused to poop outside when my daughter took her, now she also refuses with me. She will hold it for most of a day and wait for the moment when our backs are turned to poop inside, sometimes while she is eating, as she's not crated then (after all the experts say crate training is supposed to be some kind of magic bullet, so we're crating her often).

If I see her circling or if she starts making a poop inside, and pick her up and take her outside, she will stop pooping and hold it, no matter how long we're out for. If then she's put in the crate afterwards without having completed her poop, and so I know she needs to go, and it's been a few hours, and I take her outside, she will not poop. We're outside with her at least 15 minutes a go. Neither of us are frightening. My daughter is the gentlest and sweetest person who ever lived (though even she's getting frustrated). I can't just stay out with the puppy for hours--I'm a single mom and I have a household to run and a job to do, my daughter has mobility issues and requires some extra care, not to mention it's still pretty much winter where we live. Besides which, given how long she can hold it, we'd probably be out with her for many hours if we were to go that route.

Last night she was in the crate for 4-5 hours, she woke up around 1:30. Up until now at that point I've just been opening the crate to let her roam in the pen for the rest of the night, but you know, the experts say to take them outside for a brief quiet overnight pee/poo break. She peed just fine, as always. She did not poop. I brought her in after 10 ... (more)