I need to hide a conversation or reorder conversations so that the ones which might upset a customer don't appear when I am with that customer; that is, they may see I have been texting a competitor they do not like and then cancel my contract even though the texting has no impact on their business or have anything to do with them whatsoever.

Yes! One thing hangouts does better is managing a clean main page; a feature sorely missed in textra :)