I would like to create an InfoPath form that submits two fields to one list, and two fields to another list. Could I use a button with rules that bind to multiple data connections and then promote only certain fields in each connection?
This can only be done with code. You'd either need to write it yourself or use a tool called qRules from Qdabra Software. It's like an add-on for InfoPath that provides functionality that is not available out of the box. It's not free, but it costs less than a day's developer time.