It turns out that IEEE Xplore Nuclear ScienceIEEE Xplore Nuclear Science published a paper on the subject of solar cell degradation in the Van Allen belts. While I can't read the paper, it mentions that N-P solar cells (The most common) are what is susceptible to any damage by being in this region in the abstract. This paper was published in 1963. In any case, the most damaging part of the belt is the inner belt, which ranges from 600 to 3700 miles, which, correctly, a satellite going from LEO to Lunar orbit would spend significant time in.
In fact, there are spacecraft that fly in this region. The most high profile is the Van Allen Probes, which were intended to study the region. I can't actually find anything about the solar panels being a concern.
My thoughts is that this would degrade the overall lifetime of the panels, but wouldn't significantly affect the capacity. At least 2 years of operating in the worst part of the Van Allen belts is possible, and I believe this would work for some period of time, as only a relatively small portion of the mission would be in the belts. A single cargo mission? Well, there's been plenty of missions that have gone through the belts using ion drives, I don't think that's even a major concern.
