The easiest way to run an existing background segmentation effect with a face effect is to edit the background segmentation effect as shown in the screenshots.
Set the Layer to 0 for both the background and foreground
Set the Depth to 1 for the foreground to render it in front of the background
Mirror the background and foreground objects on the X and Y axis because Layer 0 is oriented differently from the Overlay layers.
This will cause face effects that are typically in layers 1-10 to be rendered over the background. Keep in mind that effects that use camera texture such as beauty effects will likely also require a render texture as an input.
The background segmentation Template in the new Studio beta is already setup as described in this post and can be used as-is.