United has a non-stop to Chicago from Dublin.
Currently Dep 12:50p ARR 3:15p
Then an evening non-stop to Bozeman - you is lucky.
Dep 6:10P Arr 8:18
(You would need to sleep as soon as you arrived.)

That would give you three hours for customs and flight change.
The Bozeman flights are seasonal in summer for tourism and winter for skiing.
The do not always show up on flight search websites.
I cannot say whether they will be offered next summer - but probably.

Total flight time - 8 hrs to Chicago, 3 hours to Bozeman - 11 hours plus 3 hrs layover.

You couldn't ask for a better schedule.