We have not had any specific reports regarding Exodus or Kodi. They may have updated it to detect programs like Jaksta and shut down the stream. We will make a note of it, but for now you may just need to wait for a future update that will hopefully address the issue.
As an alternative you should be able to use the Screen Recorder to live-capture the video.
Is there a known problem with recording from Exodus on Kodi? I've been using it to download films for a while but, since the v17 update to Kodi, Exodus won't play if Jaksta is running. Is there anything you can do to get around this? I've just updated to the latest version of Jaksta but that hasn't helped. Thanks.