I don’t know what KACE is, but it seems a little odd that your machines are dropping their network resources randomly. I’d look into your Service Options on the FOG Page and check that there isn’t a task set for rebooting the machines. This could be under the Green FOG settings or Task Reboot. I don’t know which one it will be. However as you state it only seems to happen when the WIFI is enabled, I’d check that the primary address for the hosts is the LAN not WIFI. If it’s not these options, it’s highly unlikely it’s the FOG Client causing the issue, but seemingly an issue with the image itself.
I hope this is somewhat helpful in narrowing down the issue for you.