Name a screenshot immediately and save to location other than the default?

You could create a Folder Action that asks for a name for the screenshot after it is saved into the destination folder.

Add a Run AppleScript action with this code:

on run {input, parameters}
	tell application "Finder"
		activate
		repeat with anItem in input
			
			if name of (anItem) begins with "Screen Shot" then
				set extension to "." & name extension of (anItem)
				display dialog "Enter new name for " & name of (anItem) & ":" default answer "" buttons {"Cancel", "Rename"} default button "Rename" cancel button "Cancel" with title "Rename Screen Shot" giving up after 30
				set newName to text returned of the result
				if newName is not "" then
					set name of anItem to newName & extension
				end if
				
			end if
		end repeat
	end tell
end run

If you use that folder for other things, put a Filter Finder Items action before the AppleScript action and set it to name contains Screen Shot.

Press command shift 5, click "Options" select "Other Location…".