I'd recommend looking into DesktopInfo. It takes a bit time to configure, but you can display almost anything you want. PM me if you want my .inf file that is configured to show the username, hostname and ip-address of the client. You could deploy the wallpaper with a GPO as DesktopInfo is not linked to your wallpaper.
We use this tool, Desktop Info.
This is a post on how to configure and deploy it via GPO. https://sys-advisor.com/en/2017/07/18/tuto-desktop-info-how-to-configure-and-deploy-desktop-info-by-gpo/
Personally I prefer Desktop Info over BGInfo, but similar idea - you can create a scheduled task to start it at user login https://www.glenn.delahoy.com/desktopinfo/