Axeman -
Try this (read carefully).
First, prepare your external USB drive.
Erase it to Mac OS extended with journaling enabled, GUID partition format.
Let it be for the moment (connected, but not doing anything).
Now, turn your attention to getting your Mac booted to "the right recovery mode".
I believe you want to power down, ALL THE WAY OFF.
Now, reboot and IMMEDIATELY hold down the following key combination and KEEP HOLDING IT DOWN for a while:
Shift-command-OPTION-R
(yes, I realize it's a "finger contortion")
What this should do is boot you into a recovery mode that will install the version of the Mac OS that came pre-installed on your Mac.
Hopefully, this was NOT Catalina.
IMPORTANT:
Of course, you DO NOT want to install this to your internal drive.
So....
Begin the install procedure, but watch carefully for the moment the installer gives you the option to "show all disks".
NOW you want to "point the installer" towards your USB drive, so that the OS install will go onto it.
Are you able to do this?
If so, go through with the install.
When the install is done, you should now be booted to the USB drive (in Mojave).
Go through a basic setup, get it set up the way you want.
That's how I'd try to do it.