Temperature variations and altitude acclimation.
You Brits are not used to 50F swings in temp and 8000 ft.
With a solid 2 weeks of cycling time, I would also encourage a loop into Grand Teton NP.
More than likely counter-clockwise - heading back thru Togwotee and Wind River Canyon.
Or, you can fly into Great Falls and do a loop of Glacier and Yellowstone - - easily.
Billings can easily be 35C in August - and it's a wasteland of traffic and box stores.
(But the view from the airport on the Rim is stunning.)
Cycling out of and into Billings is no picnic - others cities are easier.
Considering that you will have a significant lower altitude section to start and end with -
Late June may be better then late August. You Brits melt like buttah, eh?
PS - "Labor Day" - And Americans have no neighbours nor any appreciation of colour.