Distant object won't render in 'distance camera'

I'm making a space exploration game which deals with huge distances, so to avoid that annoying render-flickering caused by not having enough world-space z-distance 'resolution', I've split my render between 2 cameras; one for objects 0.1 to 1000 units, the other for objects greater than 1000 units distant. This mostly works fine...

....Apart from the 'sun' in my game. I've written a custom surface shader for my sun object, which has white fade to 0 alpha at the edge of the sphere to simulate a glow. The problem is, when alpha is included in the shader, the sun fails to render in the 'distance camera'. As a test, I tried extending the 'close' camera's far plane to match the distance camera and the sun - with fading edges, as intended - rendered fine.

What is it about my near camera that lets it 'see' the sun, but my distance camera doesn't? I have not set up any layers in the game, so that isn't the problem. The only difference I can see is that the near camera's 'clear flags' parameter is set to 'Depth only' and the distance camera's is set to 'Skybox'. When I set the distance camera to 'Depth only' the sun renders alright, but nothing gets cleared each frame so the screen rapidly looks like the end of 2001: A Space Odyssey (very psychedelic).

So, is there some way of setting the distance camera so it displays the Skybox and renders the sun correctly?

Here's the shader for my sun. I have tried all manner of Tags and ZWrite, ZTest settings. If anyone has any ideas, I'd appreciate it.

 Shader "Unlit/sun shader"
 {
     Properties {
       _Color ("Color", Color) = (1,1,1,1)
     }
     SubShader {
 
       CGPROGRAM
       #pragma surface surf Lambert alpha
       struct Input {
         float3 viewDir;
       };
       float4 _Color;
       void surf (Input IN, inout SurfaceOutput o) {
           half rim = saturate(dot (normalize(IN.viewDir), o.Normal));
           rim = (rim * rim) * 2;
         o.Albedo = _Color.rgb;
         o.Alpha = rim;
         o.Emission = 1;
       }
       ENDCG
     } 
     Fallback "Diffuse"
   }