For a significant part of their orbit the ISS, or satellites , are not going to cast a shadow that would fall on the surface of the earth. When they do, a significant portion of that time would result in a shadow that is very diffused. Given the distances involved I suspect it would be diffused to the point of invisibility.
I think it’s fairly safe to say that the only satellite of Earth big enough to cast a visible shadow is the moon during a solar eclipse. It’s still moving pretty quickly. It’s just that its size is enough that the shadow covers a significant portion of the Earth’s surface. Thus, moon is only seen in solar eclipse.
I hope you found this helpful, let me know by recommending this answer.
To help answer this question I would recommend getting a stargazing mobile application that can track artificial satellites. Then take it out on any moonful or moonless night and see if you can see what the app tells you is there.
A lunar eclipse would reduce the amount of light striking the dark side of the earth, thus reducing the amount of diffuse light in the atmosphere. So if an artificial satellite would otherwise be visible without the lunar eclipse, it should be even more visible during the eclipse. The earth's shadow is there anyway, whether or not the moon is in it.
Keep in mind also that the visibility of an artificial satellite is sensitive to its orientation at a particular moment. Some satellites might normally be hard to see except, for example, when the orientation of one or more of their solar panels pass through a narrow window of high reflectivity with respect to your location on the ground.
In summary, the only reason I would expect artificial satellites to be less visible during a lunar eclipse is because everyone is looking at the eclipse ;) Seriously, though, the eclipse should only help and not hinder their visibility if that's what you're looking for.