Use "startosinstall" to install macOS Catalina on External Drive
The Apple tool targets the current booted OS as far as I've been able to accomplish. Using asr is still valid, but can take some time to get the arguments correct. I think the tool you want for scripting installers is Mac Deploy Stick. You can use it without the hardware, but the hardware is quite amazing and a real time saver based on how many operations you need to handle.

https://twocanoes.com/products/mac/mac-deploy-stick/

This Sabrent gear also has served us well with push button cloning. There are many like it but this brand works for me. You can go bigger as well, based on how much time you save.

https://www.amazon.com/Sabrent-External-Duplicator-Function-EC-HD2B/dp/

You might experiment with running the installer once and then when the system boots to the external drive, instead of running the assistant, shut down the system and try cloning the drive.