$300 is what your D’s college charges to get it on the transcript. If she is getting paid (well) and can also get credit, I would pay the fee. With additional credits, your D may be able to take a lighter load later on or can afford to drop a course in case of bad grade.
Most summer internships run between 6-8 weeks, and most college students have up to 12 weeks off in the summer. We normally take our family vacation beginning of Jun when our kids get out of school, and they do their physical tune ups in Aug before they go back to school. In the last few summers, D2 has been working in the same city where we live, so she has been able to take few hours off here and there for doctor’s appts.