First off, blit should not be used with the same target for both source and destination. But the actual issue here is depth not getting bound properly. This happens because using an overlay camera causes URP to create a depth texture rather than using one that is implicitly created with the backbuffer, and that makes it incompatible with Blit. This is because Blit has no way to specify a separate depth texture, but will only use one if it's implicitly created with the source texture.

In this case, the supplied StencilTestShader doesn't read from the source, and so it does not have to use Blit. Instead, you can draw a full-screen triangle like this:
var camera = renderingData.cameraData.camera;
cmd.SetViewProjectionMatrices(Matrix4x4.identity, Matrix4x4.identity);
cmd.DrawMesh(RenderingUtils.fullscreenMesh, Matrix4x4.identity, stencilTestMaterial);
cmd.SetViewProjectionMatrices(camera.worldToCameraMatrix, camera.projectionMatrix);
This doesn't touch the currently bound render targets, so the stencil buffer stays bound, and makes the post-process effect work correctly.