There's always the possibility of more than one François born in the Jomini family about the same time, so the odds of a successful search in Payerne are increased if you can locate the exact date of birth of your immigrant ancestor from US records. Tombstones, obituaries, and naturalization records are where this information usually turns up. If he died late enough, there may be an death certificate. Also, emigrants from Switzerland usually had some sort of identity documents with them, which are often kept in the family. Check all available sources!
Given names in the Swiss records were treated with some flexibility. A baby would frequently be baptised with several given names, but the order of the names might be listed differently in later records. Occasionally the person would be known by a completely different name as an adult. Be alert for any records in the US that show your Francis with an additional given name.
Good luck with your search!