Lines within polygon selection using NeighborFinder

I need to select the lines that fall entirely within a specific polygon and update their data with the respective polygon name. I attempted to use a spatial filter, but it didn't work as expected. When a line intersects two polygons, the filter selects and attributes the line to only one of the polygons, which is causing issues. I also tried using the NeighborFinder without success. I've attached a screenshot to illustrate a line segment that should not be selected within this polygon.