Hmm. There are a couple of possibilities here to get this to work. One is, you can look at deploying an offline policy from Jamf Pro that triggers on network state change that runs a script, determines if the Mac is able to see your network, and if not, runs some commands to add the app/script into the user's Dock via something like dockutil.
The second way I can think of doing this would be with a LaunchAgent that also triggers on network state change and does essentially the same things as above - checks for your network and either exits if it can see it, or drops the utility into their Dock if it cannot.
I have to ask though, why not simply deploy it to their Dock and keep it there? Is there some reason to only place it there when they are off the network as opposed to on it?
Outside of these, you may want to consider also looking at items like custom menu bar items. There are a few tools out there that can let you create menu bar items that can do a variety of different things, including running the diagnostic commands you mentioned. It should be easy to create one that alerts the user when the network connection has dropped and changes the status menu icon to indicate as such.
