The lines are not actually disappearing.
Write this whole thing off to the limitations of the way texture technology is implemented in the software.
Imagine the vector outline as if it were a stencil and the texture line as the area you sprayed paint into. When you use the Perspective Tool you move the stencil but the sprayed paint does not move or change shape.
When you use the Perspective Tool the textured line remains untouched. What you move is the vector outline without its content. What you end up with is a vector outline that is not lined up with the content. This becomes obvious if you make small shifts using the Perspective Tool with this concept in mind.
A vector line is relatively simple. When a texture line goes down I think it is extra data that modifies the way we perceive solid black. That modification data is inflexible to a great extent and it is independent of the vector data that is changeable in the software. It looks good enough when you lay a line down the first time. If you modify a texture line after it is initially created you mess with the illusion. The texture data does not extend to everything you do to the vector line.
It has to do with the way texture is faked. You see other limitations with this faked texture technology. When erasing a textured line the Eraser Tool does not erase as though it also reads the texture data. Instead you get sharp edges and not textured looking edges. The eraser acts like it is erasing a smooth surface.
The only way to retain that textured look when erasing a textured line is to soften the edge of the eraser and change its transparency (using soft eraser). But this only helps prevent the eraser from completely obliterating the original texture data. There is no eraser tool setting that behaves like it is erasing on a textured surface. If there was you would see texture options in the Tool Properties window when the Eraser Tool was selected.
I don’t understand this enough to point you in a different direction to achieve what you are seeking. My explanation has been derived though observation and experimentation.