Preserving fine line art (variable zoom and resolutions)?

The game I'm working on uses an illustration art style with lots of thin, detailed lineart created in vector graphics tools. It looks fantastic on very high resolution displays, but at low resolutions, when the camera zooms out, the lines start flickering or disappearing when objects move around, which makes the game look ugly. How can I prevent this from happening?

I've created a simple test scene with a circle containing very thin perpendicular lines to demonstrate the issue. This is the 256x256px texture I'm using as a test:

Texture import settings:

• Texture Type: Sprite (2D\uGUI)

• Sprite Mode: Single

• Pixels To Units: 100

• Filter Mode: Bilinear

• Max Size: 1024

• Format: Truecolor

Here is a portion of a 1024x768px screen with two circles next to each other. The orthographic camera has a size of 30. This is what I would expect from point filtering on textures, but it is set to Bilinear. Is this a bug in Unity, or is there something happening that I don't understand?

After trying various strategies for eliminating the flickering and making the lines crisper, I found a script on the oculus forums (https://developer.oculusvr.com/forums/viewtopic.php?f=37&t=1738☆t=20) that implements supersampling by rendering the scene to a render texture twice the size of the screen and displaying that in a new camera that culls everything else. The result (above) is much better, but is still quite blurry because of the bilinear filtering on the render texture. Switching it to point filtering produces results identical to the original image. Why does supersampling help in this case (since the circle textures are set to Bilinear filtering already)?

If supersampling is really the best way of tackling this problem, what is the best way to downsample the large render texture using a different algorithm (not bilinear filtering)?

The camera that renders the large render texture has the script

 using UnityEngine;
 using System.Collections;
 
 public class TextureRenderCamera : MonoBehaviour
 {
     public RenderTexture CameraRenderTexture;
 
     void  OnRenderImage (RenderTexture source, RenderTexture destination)
     {    
         // Just draw the texture
         Graphics.Blit (CameraRenderTexture, destination);
     }
 }

Can I change Graphics.Blit (CameraRenderTexture, destination); to Graphics.Blit (CameraRenderTexture, destination,mat); where mat would have a custom fragment shader (possibly based on http://stackoverflow.com/questions/13501081/efficient-bicubic-filtering-code-in-glsl) applied to it? I've never dealt with shaders before, so I don't understand the pipeline yet.