I’m gonna pull a Stackexchange-style self-answer here, because I’ve figured it out
The solution is to use BackBufferCopy with the copy_mode set to Viewport and set its z_index above what you want coming through the shader reading the SCREEN_TEXTURE.
For example, here I am using a BackBufferCopy with its own z_index set to 41, so it’s above Icon2 (z_index = 40), and below Gas2 (z_index = 50).