Mesh disappears at different distances due to shader

In this scene, I've placed a mesh prefab on top of a sphere. At different distances the mesh will disappear and reappear. This happens in both the scene view and game mode. The below gif shows a camera moving forward towards the sphere in the middle and the circular red mesh blinking in and out of existence at different intervals.

My knowledge on shaders is basically nonexistent. This shader was taken from someone else on this forum because it has an alpha mask in the form of an image. I've tried adding "Cull Off" and "ZTest Off" to debug, but both return the same result. I know this is a problem with this shader specifically because other shaders don't have this "blinking" effect (in other words, other shaders are always visible).

Below is the shader code.

 Shader "Alpha Mask/Bumped Specular" {
     Properties{
         _Color("Color", Color) = (1,1,1,1)
         _SpecCol("Specular Color", Color) = (1,1,1,1)
         _Spec("Specularity", Range(0,1)) = 0.8
         _MainTex("Albedo (RGB)", 2D) = "white" {}
         _BumpMap("Normal Map", 2D) = "bump" {}
         _Mask("Mask (A)", 2D) = "white" {}
     }
 
     SubShader{
         Tags { "RenderType" = "Opaque" }
         LOD 200
 
         CGPROGRAM
         #pragma surface surf Spec fullforwardshadows addshadow alpha
 
         // Use shader model 3.0 target, to get nicer looking lighting
         #pragma target 3.0
 
         sampler2D _MainTex;
         sampler2D _BumpMap;
         sampler2D _Mask;
 
         struct Input {
             float2 uv_MainTex;
             float2 uv_BumpMap;
             float2 uv_Mask;
         };
 
         half _Spec;
         fixed4 _SpecCol;
         fixed4 _Color;
 
         half4 LightingSpec(SurfaceOutput s, half3 lightDir, half3 viewDir, half atten) {
             s.Normal = normalize(s.Normal);
 
             half3 h = normalize(lightDir + viewDir);
 
             half diff = max(0, dot(s.Normal, lightDir));
 
             float nh = max(0, dot(s.Normal, h));
             float spec = pow(nh, 4096.0 * pow(s.Specular, 2)) * s.Specular * 2;
 
             half4 c;
             c.rgb = (s.Albedo * _LightColor0.rgb * diff + _LightColor0.rgb * spec * _SpecCol) * atten;
             c.a = s.Alpha;
             return c;
         }
 
         void surf(Input IN, inout SurfaceOutput o) {
             // Albedo comes from a texture tinted by color
             fixed4 c = tex2D(_MainTex, IN.uv_MainTex) * _Color;
             o.Emission = c.rgb;
             o.Specular = _Spec;
             o.Normal = UnpackNormal(tex2D(_BumpMap, IN.uv_BumpMap));
             o.Alpha = tex2D(_Mask, IN.uv_Mask).a;
         }
         ENDCG
     }
     FallBack "Diffuse"
 }

Any help is appreciated, thanks in advance :)