It is not obvious that Forge is incapable of actually saving your map while offline, because:
You can still access Forge while offline, or if you set your server in Custom Games to “Local Splitscreen” while online
You can still attempt to save your maps, although saving actually does nothing, not even saving a copy to your device when prompted to give your already named map a new one when attempting to save from the Pause Menu
If you try to circumvent saving from the Pause Menu, and do it through the quick menu (holding X), the game tells you that the map has been saved. But I have nothing to show for it. I spent half an hour trying to find the map I saved, to no avail
Please let local copies be saved, or allow (re)connecting to the server from the Pause Menu, or check for online connection and give out warnings before starting. Also, don’t let Custom Games server connection settings mess with Forge, because that’s the only reason I’ve deduced me starting Forge offline; right after exiting the Custom Game menu while having the connection set to “Local Splitscreen”.