As a historian of the American West and a cyclist with tens of thousands of miles of riding in the West, I can say that you, like most other Brits, have little sense of the scale of the region. It would take a good three weeks to ride from Seattle to San Francisco - your proposed direction is entirely wrong because of prevailing winds. Seattle to Denver would take four weeks at the very minimum. Also, the month makes a big difference - not just "summer". June may be too early to ride certain areas - while August would be brutal in the Southwest. You might be able to ride from San Francisco directly to Denver in four weeks, but that would be a tough slog for the uninitiated - 10,000 feet of climbing in the Sierras by the 3rd or 4th day and remote riding with zero services for 80 miles at a time across the Nevada desert. Remember - less is more.

A tour starting in Seattle or Portland to Glacier NP and then down to Yellowstone NP is more doable in mid-summer.
From Jackson Hole there are flights to majro international airports.