Before we dive into what might be helpful, I want to take a moment to honor your deep commitment to helping your BF’s daughter. This is important because beyond the immediacy of the solution to this issue there will be the love and caring that you are offering to help her through this. More than any technique your deep desire to help will be what endures.
That being said, the natural disruption in the routine makes sense for throwing off her sleep. While it is hard to know the first thing that is usually recommended is to extend whatever is working. In this instance, I would have your boyfriend record the reading of the Harry Potter chapters. When she wakes up there will be a natural remedy and a reminder. Allowing her the ability to drift back off to sleep while hearing her dad’s voice may be just the antidote to the sleeping issue. This may be the cue that reminds her of the natural pattern to return to sleep. This extends the comfort of the ritual by replicating dad’s voice.
Second to this is to keep as much of what is a routine — a routine. If she is used to half a weekend then maybe keeping what is possible of that weekend possible may be of help. In other words, because of the Saturday work schedule the Friday was skipped for the every-other-weekend. I’d experiment with keeping the Friday nights and looking for an alternative to returning her on Saturday. Having such a long time between connection may be part of the issue — and keeping the Friday time — even if it is limited or has a change in the return process for Saturday may help to stabilize the situation. Again, often what remains isn’t the exact solution to the issues, but rather the loving and caring intention you are extending.