Hi Barry,
I am not familiar with home assistant and I have no idea how the home assistant can manage this.
If I would need to implement the clock synchronization only, I would probably create a StartThings SmartApp and hosted it on Amazon as a AWS Lambda. (see Building First SmartApp for Dummies).
It should be a very simple stateless app. Each time it wakes up by timer, it enumerates all devices and sends the execute commands to devices with clocks as @Simon4 described above.
In this case, if you want to update clocks, you install the app; if you want to stop it, you uninstall the app. No UI, no data to persist. Easy.