live share will always create a invitation link upon starting, so you can use clipboard listener based on event listeners or pyperclip (repeatedly checking content in clipboard, see if there’s any change), save the link to server or send via email.
if clipboard is not your thing, there’s a massive anyIM-to-Matrix list
you can also setup your own sync app via instant messaging services.
clipboard sync:
crossclip: Sync clipboard across macOS/Linux/Windows on LAN
ClipBroad: Sync via Github
clipboard listeners:
clipnotify for xorg (linux)
windows+v now act as multi-clipboard
you can sync clipboard across computers via crossclip
microsoft liveshare