Doom-style billard/sprite skybox without perspective warping?

It's not entirely clear what you actually want. The original Doom doesn't allow looking up or down. It uses a cylindrical projection of the skytexture. Later there where free look mods which only applied "Y-shearing". So they did not rotate the camera but simply shift / shear the projection along the up axis. That way the actual scene was still rendered completely horizonal.

There are other doom remakes which offers actual freelook. They usually use a spherical mapping of the skytexture or use an actual skybox with 6 textures.

Note that for the cylindrical or spherical mapping you do not actually render a huge quad in the background but you simply use a shader on the sky brush which does the coordinate mapping based on the camera direction. This is what a spherical mapping looks like:

There are just two quads with my DoomSky material which uses the DoomSky shader i just created:

 Shader "FX/DoomSky"
 {
     Properties
     {
         _MainTex ("Texture", 2D) = "white" {}
         _StretchDown ("Stretch", Range(0, 0.5)) = 0
     }
     SubShader
     {
         Tags { "RenderType"="Opaque" }
         Cull off ZWrite On ZTest Lequal
 
         LOD 100
 
         Pass
         {
             CGPROGRAM
             #pragma vertex vert
             #pragma fragment frag
             
             #include "UnityCG.cginc"
 
             struct appdata
             {
                 float4 vertex : POSITION;
             };
 
             struct v2f
             {
                 float3 worldView : TEXCOORD1;
                 float4 vertex : SV_POSITION;
             };
 
             v2f vert (appdata v)
             {
                 v2f o;
                 o.vertex = UnityObjectToClipPos(v.vertex);
                 o.worldView = -WorldSpaceViewDir (v.vertex);
                 return o;
             }
             sampler2D _MainTex;
             float _StretchDown;
             
             fixed4 frag (v2f i) : SV_Target
             {
                 float3 dir = normalize(i.worldView);
                 float2 uv = float2(0.5+atan2(dir.z, dir.x)/(2*UNITY_PI), _StretchDown+dir.y*(1-_StretchDown));
                 fixed4 col = tex2D(_MainTex, uv);
                 return col;
             }
             ENDCG
         }
     }
 }

Note that the texture's wrapmode is set here to "repeat". That's why you see the same texture below the horizon. Originally it only ever reaches down to the horizon. Also note that since the textures were never meant to be used in an actual free look environment (even only considering the upper hemisphere) the pole right above you will look strange as, just like every spherical projection, the top edge of the texture will collapse to a single point. The SkyTextures where only tiling horizontally so the left end matches the right one.

I've added a "Stretch" parameter which allows you to move the horizon down a bit. It will stretch the upper hemisphere so it reaches further down. While a value of 0 means the horizon equals the bottom of the texture, a value of 0.5 means the texture will be stretched across the whole sphere. So the top edge collapses at the north pole while the bottom edge would collapse at the south pole.

Generally if you use this technique you either want to restrict the cam to only rotate around y (just like the original doom) or make sure your texture properly maps to a sphere. For example a normal equirectangular map works just fine with stretch set to 0.5. I've tried it with an earth texture and i get the earth mapped as sky sphere around and it looks great in all directions, even the poles.