I do not believe what you're asking is going to be easily doable. I believe the secondary cut and paste functions are isolated in each app that supports them and it would probably require programmatically trapping the target key sequence and processing accordingly. This is something that would probably need to be written in C and or Objective-C, compiled to executable and stay memory resident.
That said though, I believe that the ^K shortcut only cuts the last line of text from the position of the cursor and places it on the secondary pasteboard. So with that as the understanding, let me offer the following as a workaround solution.
The following example AppleScript code, shown further below, will be used in an Automator service, where one can then assigned a keyboard shortcut to the service. This service will use e.g. ⌃⇧K (Control-Shift-K) and achieve the goal of the OP. Which is to get what is placed on the secondary pasteboard in Terminal, by normally pressing ⌃K, onto the general system pasteboard a.k.a. the Clipboard, albeit with pressing e.g. ⌃⇧K instead.
The following was created, tested and works for me in macOS High Sierra.
Create the Automator service...
In Automator:
- File > New, or press: ⌘N
- Select Service and click the Choose button.
- Configure the settings as shown in the first image below.
- Add a Run AppleScript action.
- Replace the default code with the example AppleScript code.
- Save as e.g.: My Ctrl-K for Terminal

Example AppleScript code:
tell application "Terminal" if (count windows) is greater than 0 then try activate set c1 to contents of selected tab of front window tell application "System Events" to ¬ keystroke "k" using control down delay 0.5 set c2 to contents of selected tab of front window end try else return end if end tell tell current application set pCount to (count paragraphs of text of c1) repeat with i from 1 to pCount if (count characters of (paragraph i of c1)) is greater than 0 then set t1 to (paragraph i of c1) set t2 to (paragraph i of c2) end if end repeat if (count characters of t1) is not equal to (count characters of t2) then set AppleScript's text item delimiters to {t2} set CtrlY to text items of t1 set AppleScript's text item delimiters to {} set the clipboard to CtrlY as text end if end tell
- Note: The example AppleScript code is just that and does not contain any other error handling then what may be present. The onus is always upon the user to employ proper error handling as necessary and or wanted.
- The value of the
delay command may or may not need to be adjusted for your system. Adjust the value as may be needed and or add additional delay commands as may be needed.
Assigning the keyboard shortcut...
In System Preferences...

Now in Terminal, when one presses ⌃⇧K you'll get on the Clipboard that which ⌃K places on the secondary pasteboard.
As a side note, if you use a program like FastScripts, you only need to use the example AppleScript code as a .scpt in Script Editor, not create an Automator service and can assigned the keyboard shortcut in the Preferences for FastScripts. None of the other instructions apply unless you want to do it all natively without the use of third-party software.
Note that I am not affiliated with the developer of FastScripts, just a satisfied user.