System App Permissions
As ProNetGuru already pointed out, Titanium Backup could indeed be used for that:


root
install TiBu
convert app to system app (i.e. it gets installed to /system/apps
uninstall TiBu
unroot.


Done. That app would have access up to "signatureOrSystem" protection levels then (if specified so in its Manifest).

Giving it root access is something different – but probably not even needed here.