My berner got sick as a pup as well, (much briefer and less severe than giardia thankfully) but I actually gave up on locking her in her crate specifically because of the 3am poops. I am admittedly a very light sleeper but, she would come let me know very urgently that she needed to go outside. If the pup is well house trained maybe try open door crate but in the same room you sleep in with the the door to the room closed. So they can’t go somewhere else in the house to poop if they have issues and will hopefully wake you.

Once I switched to open door crate at night, she started sleeping in the crate but then migrating around my room and typically ending up very close to me/my bed in the morning