Installing Mojave on an external drive

I have: an iMac running Catalina 10.15.2. and two external hard drives: a G-Drive mobile and a WD My Book. 

What I want to do

I use the G-Drive in conjunction with Time Machine so everything on the iMac is safely backed up. While retaining Catalina on the iMac, I want to install Mojave on the WD so that I can use software which is not Catalina-compatible. I am part of the way there. I have successfully created a bootable Mojave installer. It is on a flash drive which has changed its name to Install macOS Mojave. And I have erased the content of the WD and renamed it My Book Mojave. Its icon on the screen now also shows as My Book Mojave. But now I hit problems.

(1) If I turn on the computer while holding down the Option key, I first get what looks like a line drawing of a mouse alternating with a line drawing of something else, and the cursor is frozen in the top left-hand corner of the screen. If I hit the space bar this gives way to the macOS Recovery screen. I can navigate between the four options using the space bar but cannot click the Continue button as the cursor is still frozen. (2) Trying a different tack, I boot up holding down Command + R. This takes me straight to the Recovery screen and this time the cursor is not frozen. But when I click Install OS, it wants to reinstall Catalina on my iMac with no way of choosing a different OS or destination. (3) In Startup Disk in System Preferences only my iMac hard drive is shown. Even If I could find a way of showing the Mojave Installer there, wouldn’t it install it on my iMac hard drive? (4) If I double-click the Mojave Installer icon on my screen, it first says it’s verifying and then informs me that this version of the Installer is too old to be opened. 

Can anyone guide me in the right direction?