And there you have it - alpha clipping no longer works in the custom pass.
I have tried all possible combinations of Injection Point and the various other dropdowns available in the custom pass settings. Alpha clipping never works.
I believe this is because in the HDRP Lit shader, for opaque objects we do the alpha clipping during the depth pre-pass and then we disable it for performance reasons.
In this situation, I think the best would be to add another draw renderers custom pass executed in the “After Depth And Normal” injection point that writes the depth of your alpha-clipped objects into HDRP depth buffer. Then you can disable the Override Depth toggle in the Draw Renderers custom pass.
I think you can take a look at this example from the custom pass repo: https://github.com/alelievr/HDRP-Custom-Passes#render-with-normal-buffer (Rendering an object with multiple custom passes in depth, normal and color buffers).
Rendering an object in such a way will fix the alpha clip problems as well as other issues related to post processes.