How do I determine an external drive's macOS version?

In AppleScript, without having to launch Disk Utility this will allow you to pick what is in the /Volumes folder (in particular, a mounted external drive) and display the information shown in the dialog below. Just copy/paste into the Script Editor, click the compile button, and run… I don't have an external bootable drive with me (traveling) to test that functionality.

use framework "Foundation"
use AppleScript version "2.4" -- yosemite or later
use scripting additions

property NSDictionary : a reference to current application's NSDictionary
property SysVersion : "Macintosh HD/System/Library/CoreServices/SystemVersion.plist"

set VolPath to POSIX path of (choose folder default location (POSIX file "/Volumes" as alias)) as text

if VolPath = "/" then
	set fullPath to (VolPath & text 14 thru -1 of SysVersion)
else
	set fullPath to POSIX path of (VolPath & SysVersion) as text
end if

set dict to NSDictionary's dictionaryWithContentsOfFile:fullPath
set theOS to (dict's valueForKey:"ProductUserVisibleVersion") as text
set buildVersion to (dict's valueForKey:"ProductBuildVersion") as text
display dialog "macOS " & theOS & " (" & buildVersion & ")" with title "macOS Drive Version"
return

Open Disk Utility, and select the drive.

I don't know the answer to your question but, did you allow external booting in Startup Security Utility?

About Startup Security Utility on a Mac with the Apple T2 Security Chip - Apple Support

What did you use to create the bootable drive to begin with?

You won’t be able to boot a Mac studio from a boot drive created using an Intel based Mac regardless of what version it is.

+1