National Treasure/ Da Vinci Code

Short answer: Crossover! And head for Latin America. 

The first National Treasure movie deals with the American Revolution and the Civil War. 

The three Robert Langdon (aka Da Vinci Code and its sequels) movies are all set in Europe.

And there are four presidents big-deal enough to be on Mt. Rushmore-- Washington and Jefferson, who were involved in the Revolution, of course, and Lincoln presided over the Civil War.

Leaving Theodore Roosevelt. His war? The Spanish-American War. Which, you guessed it, was with Spain. 

Which is in Europe. Which is where the Da Vinci Code movies are set.

So there's the plot. It involves clues on the American side pursued by Ben Gates (Nic Cage) and Spanish clues pursued by Robert Langdon (Tom Hanks).

The Spanish-American War involved the Caribbean and also the Philippines and even Guam. So the sleuths could run into each other in any of those locales. They might first be suspicious of each other but quickly learn each is only in pursuit of truth. 

Because the plot involves Spain, any former Spanish colonies, and any history of the Inquisition-- which was run by the Catholic Church, which is Langdon's expertise-- can come into play. So can the conquistadors, lost gold and other artifacts, etc.

Because Teddy Roosevelt was involved, other locations come into play. He founded the National Parks system, for instance. He worked as a rancher out West, and for the police force in New York. He was McKinley's Navy Secretary, and later hunted game in Africa, among many other places his amazing life took him. 

One was the Panama Canal, which he pushed for, and that is also in former Spanish colonial land. 

The clues could lead all over the world, necessitating both the skills of Franklin and Langdon, as well as their teams. One could even be a... teddy bear. 

There are really only two other wars, fought pre-1900, well-known enough to warrant tie-ins with the National Treasure franchise: The War of 1812 and The Mexican-American War. There are other famous incidents that could be used as well: The Raid on Harper's Ferry, The Alamo, the completion of the Continental Railroad, etc. 

And then we could always go back to the American Revolution and the Civil War-- it's not like there aren't other historical figures, landmarks and documents to reference. 

Plus, who says only president and generals were involved in such doings? Certainly the great money-makers of the past-- inventors, bankers, factory owners, rail and oil barons, etc.-- were capable of hiding clues and treasures. By now, even the 1920s are "a hundred years ago." So you can add steampunk and early electrical doodads into the mix. 

Note: There is also a 3rd Langdon book, which can be turned into a TV or streaming movie.