Length contraction at an Angle 
If the ship arrives at the slit at $45^\circ$ and the slit projection onto the ship is narrower than the needed then the ship won't make through. However, if it travels at 90% of the speed of light there should be length contraction, so would it pass in this case?
 A: The dashed line through the center of the ship helps you think about this in the lab frame. It shows you all the places the nose will be. 
Add dashed lines for the edges. They show where the edges will be. They will not get closer together if the ship travels at high speed. If they cross the slit, the ship will be too wide. Speed will not help. 
You can also do the same thing in the ship frame. You have a dashed line through the center of the slit showing where the center will be. Add dashed lines to the edges of the slit. If those lines cross the ship, the slit it too narrow. 
The angle is at an angle. But the lines show the component of the width perpendicular to the motion. This component does not change. 
The component parallel to the motion will be contracted. If you draw that, you will see the slit changes angle. This is one of the counter intuitive effects of relativity. It is much the same as seeing a sphere at rest, but a flattened elliptical cross section at speed. 
A: Length contraction at near light speed, or Lorentz contraction, is effected by several other phenomena due to general relativity.  But as I understand it, it appears to any observer that the object would contract in the direction its traveling in by (1-v²/c²)½  v= relative velocity of the moving object, c= speed of light in a vacuum.
       From what I understand though, it  only appears to an observer to contract, so say it was a ship with a passenger, it wouldn't contract from their frame. Im going to say I dont think it works in you question...... But maybe this will help out.
  https://physicsworld.com/a/the-invisibility-of-length%E2%80%AFcontraction/ 
