I, too, have to use iTunes (Apple ID) rather than VPP to distribute apps. In the past I've gone about this two ways. One is to create an iTunes backup and one by one go through and restore a group of iPads. Of course, all the apps installed on the all the iPads are setup under the same Apple ID, which isn't a perfect system, but it works. The other is to use Configurator 1 to create a master backup, add the apps (from iTunes with an Apple ID) and restore a larger group of iPads all at once. This has the same effect in which all the apps are setup under the same Apple ID.
I've been using Configurator 2 and it seems to work very closely with iTunes. In fact, if you create a backup in Config 2, it actually saves in the same place that iTunes backups save. So essentially, they are exactly the same. Where as in the past, with Config 1, I would create a special file (.iosdevicebackup). Once Config 1 required that you update every device, it became troublesome. Anyway, you can restore a group of iPads through Config 2, however, the apps wont pull across. In order for apps to be pulled across onto and iPad you're looking to restore, you'll need to use a Blueprint. With a Blueprint, you can add the necessary apps two ways (I'm excluding VPP since that's not what I'm after). The first is by just adding the ipa file and the second is by signing in with an Apple ID, which gives you access to all "purchased" apps. From there you can apply this Blueprint to a group of iPads that are plugged in. The iPads will receive most things saved in the backup (wallpaper, app arrangement, certain settings, etc.) and any apps that you chose. The problem is that before opening any apps that have been pushed to recently restored devices, you will first need to enter the password to the associated Apple ID. From what I've gathered, there is no way around this. I've tried almost everything I could thing of to circumvent this but have had no success.
I know this is a long explanation and probably most people have run across this same problem. If anyone has had any success, I would be entirely grateful if they would share. I have hundreds of devices that come through my door on a weekly basis and this would save me and incredible amount of time.