Is there a way to keep track of Parent Joins in Spatial Joins? (Multiple Spatial Join)

In the attached Image, I would like to do a Spatial Join From Layer 2 to get the intersecting Layer 1’s ID. Afterwards, I would like to get all of the Layer 2 that aren’t initially spatial joined (non-parents) to connect to the one’s that are spatially joined (parent) if they intersect with a parent or another line that intersects with a parent get the ID. I was thinking of doing a spatial join intersect, and then a spatial join closest Parent; however, I think there would be limitations in the data as some of the lines could be closer to one that isn’t actually connected.

PS: If there’s also a way to get Layer 2 (all features) to have the Layer 1 ID instead, that works too (except for closest spatial join)