Thank you for your answer Ray!

You mentioned that another layer for conflict resolution will be necessary. Is there an actual hands-on example or tutorial that you could point to? 

Also, do you have any suggestions for handling multiple sites? As I mentioned, we have to know which invoice was created at which site. What entity should be handled in Odoo to make this distinction? A warehouse or Company would be a poor choice IMO.

The only parts of Odoo that work offline are Point of Sale (via the browser) and Timesheets (via the mobile Apps on iOS and Android).

Odoo has worked on setting up multiple on premise systems that can synch with each other (and as long as every user has access to the server, they don't need continuous access to the internet) but this is not trivial and we recommend purchasing a backup mobile based internet connection - like almost every ATM in the world has - because this is far cheaper and allows people to bypass all of the replication issues.

Some very skilled Customers and Partners have also setup systems like this but they leverage AT LEAST PostgreSQL replication and, often, another layer to manage situations where multiple updates need to be resolved (the customer visits one site to purchase but then another site handles an update to a phone number).

You can re-write the sequencing logic to incorporate additional factors such as the user who created the invoice.

You can get pricing on phone based internet backup connections from the following company who also happens to be an Odoo Customer - https://www.ventusgns.com/