How can I apply a Group Policy wallpaper and overlay BGInfo details on top of it?

Be careful about putting sensitive host information directly on the desktop. BGInfo itself is not automatically a security vulnerability, but displaying IP addresses, usernames, domain details, or other environment information can disclose useful data to anyone who can see the screen. Limit the fields to what support actually needs, or use a central management system instead.

It is worth questioning whether end users need this information displayed permanently. Support staff can usually retrieve the computer name and hardware details through management tools, inventory, or remote administration. If users do need a quick reference, showing only a small set of non-sensitive details is safer than exposing broad system information on every desktop.

The two settings are fighting over the desktop background. A wallpaper policy can reapply the configured image after the logon script runs, so BGInfo's generated background gets replaced. Rather than having each one set the wallpaper independently, use a script or tool that takes the standard image, adds the BGInfo text to it, saves the combined image locally, and then applies that single image as the wallpaper. You may also need to control the script's timing, but relying on logon order alone can be unreliable.

If you still want similar functionality, consider a maintained PowerShell-based alternative that can write system details onto an existing wallpaper and be launched by a logon script. The important part is that only one mechanism ultimately controls the wallpaper; the Group Policy wallpaper and BGInfo should not both try to manage it.