One of the First things the Amrican ngineer in Charge did --Was Ordered the Cut doubled in width to help slove the bank collaspes [still a major problem today]. the Second thing was to start construting a major railroad system [The Zone has more Raillines than the whole rest of Panama] to move the Earth. the Third thing was to go with a lock system, instead of the French Sea Level plan.
Both of these things the French Could of Done--Except the US was backed by the US Tax Office. While the French was a Stock Company out of Money.
But if some how the French had completed the Columbian Canal. I see a lot more Italian Immigrants into the area. with a chance that the panama highlands could be a major wine exporter today.
The French Government annexing the canal zone is unlikely to happen without a major international row. But by 1883 the USA will have no chance of forcing anything on a France with something serious to defend. Rather everybody will look to the British, who anyhow had been the real enforcers of the Monroe Doctrine since 1820.
A government like the French controlling the Panama Canal is against all British principles of free trade, and it is still not 70 years since Napoleon last popped up.
I could imagine this ending up in some kind of international mandate letting the Brtish oversee the management of the Canal, and the USA seeing this as a good alternative to the French. The Suez Canal was built by the French, but nevertheless ended up under British control. The presence of USA in relatively close proximity (and distance from Britain) will however have the British to accept a less tight control of the Panama Canal. By OTL 1914 things had changed a lot in relative strength ratios and not at least it was a canal actually built by the Americans. Seen from London it was quite a good alternative to have another "free trade freak" control the canal rather than dispersing own resources or have some unpredictable locals do it (the Egyptians were not allowed in at Suez until the British and French had long since lost their global status).