Do you need to be logged on graphical interface? (does the backup app has to be running graphically on the admin user?)
 Or just logging the admin user on background would do it? (in this case there would be no need to have the backup app opened graphically as it would just run automatically via job on the admin user, once it is logged)
 Another option would be to create a simple automatic service to run in background, for the admin user, to run automatically. (but also depends on how the backup app works)
I know how to do it the 3 ways above,
but you have to tell me more details about how the backup app works, so I can know which of the options above will serve better for your situation.
the whole macOS Mojave (and likely Catalina too) thingy is not functional to managing wake-up, sleep modes. Oh well...
i did some research into SysPreferences->UserGroups->fast user switching.
This seems to work for my needs, that is enough of the backup app is still running in the background even while another user account is active. It is difficult to get the family to use the fast user icon in the upper right of their screen. Go figure.