Is there any way, perhaps with software, that we could remove a moving object, from an otherwise mostly static set of images?
A very simple and common way is to stack all of the static images aligned on top of each other such that you have a set of values for each pixel corresponding to the same region of sky. Then sort the values of each pixel by brightness and keep the median value for that pixel. Or throw out that top and bottom 10% and average the rest, or throw out the top and bottom N, etc.
This is a standard feature of essentially all astronomical image processing software and has been for a long time.
Remember you don't need to throw away the whole frame, only the pixels that were obstructed. One track across a frame will obstruct less than 1% of the data in that frame.