Attempted to install Catalina in Target Disk Mode.... bricked my mac? Ok so I may have bricked one of my MacBook Airs.

I am writing this post from the MB that's working. I have another one with a dead display. I connected them by Thunderbolt and mounted the broken one as a disk onto the working one using Target Disk mode. Then I used the disk utility to reformat the drive as APFS, Guid Partition. Then I tried to install Catalina but it says "You may not install to this volume because it's a Mac in target disk mode" (see screenshot.)

Have I created an irrecoverable situation?

The only solution I can imagine is to remove the HDD from the bum Mac, put it into the working Mac, install Catalina that way, then put the drive back. Both Macs are out of warranty so it's whatever... but I'm wondering there's a less messy way to get myself out of this hairy situation?
What many people fail to take into account when they install a new version of macOS on a Mac, is that the process does not simply involve copying files to the Mac's internal hard drive, but also a firmware upgrade to make the firmware/EFI support booting from the new OS. This is why you get the message you get: if you boot a Mac in target mode and attempt to install a new version of macOS onto it, the Mac you install into would not boot (if you had managed), because it would never have had the firmware necessary to boot installed. 

For instance, when installing High Sierra on a Mac, the firmware necessary for the boot loader to recognise and boot from an APFS volume and disk structure must be installed, since HS was the first version of macOS to be able to boot from an AFPS volume. Similarly, when installing Catalina, the firmware necessary for the boot loader to parse the new two-volume read-only / read-write setup must be installed on the Mac that will boot the installed OS

So in short, the only viable way to be able to install (some versions of) macOS on a Mac, is to actually boot that Mac one way or another and install the version of macOS from the it, from the Mac that will run what you install.

In your case, if the only problem is a broken display, a viable solution is to boot the Mac with an external keyboard, mouse and display plugged in, while connected to a power adapter, since that should put the Mac in clamshell mode and fully operational. If you have already wiped the internal disk, put the working Mac into target mode and boot the broken Mac from the OS installed on the working Mac, plugged in via thunderbolt or Firwire (depending on connectivity options).