I disagree that my comments were not relevant.
The question at hand is whether you can maintain concentration through a short rest on a Hex spell. My comments relate to other considerations outside the concentration and rest rules that should be considered when approaching the situation.
They are relevant only to your personal campaign. You may choose to run the Patrons as petty micromanaging bureaucrats rather than powerful extraplanar entities with more important things to worry about than whether or not one of their warlocks is "wasting" their powers killing a rat, but I don't see how it's relevant to the larger discussion.
Your Patrons seem to be more concerned about the actions of their warlocks than a Deity is about the actions of their Cleric. It strikes me as silly and very much not how warlocks operate (more of a business arrangement than a master/servant or god/worshiper relationship), but it's your game so have fun with it.
I just don't see how it is relevant to a discussion of the game mechanics, or even how other people may choose to run it in their campaign.