Thanks a bunch for the saves!
I loaded up your save, progressed to day 388 when Cait is in the library in the afternoon.
Talk to her > Study with her > Sit on my lap ?
This should pop more dialogue that leads to the scene.
However, you may have to reload or retry a few times for it to work.
First visit - 33.3%
Second visit if not successful on the first - 50%
Third visit should be - 100%