Using the Startup key ***Option (⌥)*** or ***Alt*** would normally get you a menu to then select the boot drive from a list using the arrow keys of the keyboard.
So while you can’t see the list we know it’s there! So if you click on the down arrow one time you know you will get the next drive (the external in this case). Then hit the return key to accept it. That should work with your bootable OS installer plugged in.
Reference: [https://support.apple.com/en-us/102603|Mac startup key combinations]