Calmbake,
If you want to get a pound of flesh, make it a big deal.
I personally would do the following, in the most kind and gentle manner possible.
Call him and get him on the line. Tie in another elder from the hall as a witness. Then explain very kindly that his private texts to his wife are not welcome. You know he doesn't mean anything bad etc... but that you would like them to stop. It will embarrass him, you don't come off as a lunatic, and you can even bring up that section of the elders manual already shared. Be super kind. it will stop, and put him on notice that you are no dummy.