We were able to get Lola out of her cage in the early evening by using some subterfuge and a carrot. I got her scent all over me and then the son held her while I transferred her kits to the nesting box. All six look very healthy and were very squirmy and vocal. I covered them up well with her pulled fur. She had two in the far back corner and four closer together about mid-way up the side of the cage. There were no dead kits and no sign of blood or placentas, so she ate all of that up like a good Mama rabbit.
I moved the nesting box into the corner where she had had them, the furthest away from the door. I removed a good portion of the hay, but left enough in to give her a comfortable bed to rest on. Then the son put her back in the cage and we watched her until we were sure she knew where her babies were. Then we gave her another piece of carrot and she seems to have forgiven us for interfering in her cage.
Because Lola is so sensitive to cage intrusion, we will not check on her kits again until Saturday, though I will keep an eye out to make sure none of them hang on while nursing and end up out of the nesting box.