macOS Catalina has a new feature to further isolate the OS from data. In the APFS container it is installed within, an APFS volume for the OS is created alongside one for your data. The former is mounted read-only within Catalina, and the latter has the Data postfix containing your apps and data.
https://developer.apple.com/videos/play/wwdc2019/103/
Within Catalina, Disk Utility shows these volumes with a Finder icon and home icon to distinguish them. This does not appear in Mojave, showing as normal volumes. Keep in mind the release notes for the beta of Catalina, specifically dual booting and Spotlight indexing, which will heavily confuse Mojave with the cross-volume linking of various folders.
I was in contact with Apple's support for this issue, and it was ultimately resolved although a bit tortuous.
There are several modes of MacOS Recovery, accessible by holding different key combinations as described here.
Trying to access MacOS Recovery failed and gave the "-1008F" error every time I tried the MacOS Recovery modes "Reinstall the latest macOS that was installed on your Mac (recommended)" (⌘+R) and "Upgrade to the latest macOS that is compatible with your Mac." (⌥+⌘+R).
However, the mode "Reinstall the macOS that came with your Mac, or the closest version still available." (⇧+⌥+⌘+R) did work for me. From this mode, I was instructed by Apple Support to wipe/format the disk entirely clean using MacOS Recovery mode's Disk Utility and create a new volume (I'm not sure if this step is absolutely necessary). From there on, I could successfully install MacOS Mojave (the OS that originally shipped with the Mac).
Next arose the issue of restoring my system. My intention was to stay on MacOS Mojave instead of MacOS Catalina, but unfortunately it appears that when I installed MacOS Catalina the first time (before all this happened and I decided to go back to MacOS Mojave) – MacOS Catalina actually converted format of my Time Machine backups stored on my NAS from the traditional .sparsebundle format to the newly introduced .backupbundle format. I think this was disingenuous by Apple, because I had not been asked for this conversion to occur, nor did any Time Machine backup process once MacOS Catalina was installed ever move past the backup preparation stage since I had manually disabled Time Machine backup. In other words, MacOS Catalina converts your Time Machine backups to a new format without asking you!
The problem is of course that the new .backupbundle format of Time Machine backups cannot be identified by MacOS Mojave. According to Apple Support, Apple does for the time being provide a "secret" method to convert .backupbundle files back to .sparsebundle, but this is not something that they are announcing publicly because apparently they want all users to shift to MacOS Catalina. Rather than hassle with this however, I opted to upgrade to MacOS Catalina and try to restore my Time Machine backups from there instead.
Once MacOS had been upgraded to a clean install of Catalina, a had a new problem because the Time Machine backups were now identifiable but not restorable by Migration Assistant.app, the recommended Time Machine restoration application in MacOS. I believe the error message said something like "Could not open backup, file in use..." (paraphrasing). It turns out that there was an issue with MacOS/Migration Assistant not being able to open Time Machine backups stored on my Synology NAS devices over the network, even when trying different modes of SMB/AFP. The solution was to copy the .backupbundle files from the NAS onto an external USB/Thunderbolt drive, then mount the .backupbundle file by double-clicking it in Finder, and finally running Migration Assistant again. From thereon, I could successfully restore my files and system configuration onto MacOS Catalina.
The first lesson here is not to keep Time Machine backups only on NAS devices, because they are unsupported Time Machine destinations in Apple's view. Apple primarily supports Time Machine backups on external USB/Thunderbolt drives, and were not inclined to help me resolve the problem of reading the Time Machine backup on the NAS. I had to buy an entire new external drive in order to relocate the .backupbundle file and restore it, and I now intend to back up both to the external drive and NAS continually. The second lesson is to make copies of your Time Machine backups before upgrading to a new OS – I did not foresee that MacOS Catalina would convert or otherwise do things to my backup files upon OS upgrade without my permission!
Hope this is of use to anyone else running into this chain of issues after upgrading to MacOS Catalina.
Best Answer
This is pretty easy if you have a network that works or a second Mac.
I’m not sure you have either based on the initial version of your question. You may have both based on the edit.
The erase install process works great for us on the hardware you have. I wouldn't mess with bootable media if you can download from Internet Recovery.
Instead - erase the disk, which returns you to the Catalina installer. While it's running - look for Command-L and Command-3 to show the log file and show all logs.
It's unlikely you need Apple Configurator, but know you can bypass everything and start by restoring the T2 chip and then write a new BridgeOS if needed on T2 Macs.