Thanks for all your advice - much appreciated.

Each time I landed on United.com, the cheap fare was shown, then either didn't go through or jumped up when I tried to book it, depending on the device.

I tried leaving the third country in the credit card billing address and entered my US details for everything else, including the rest of address. To my surprise, it went through.

I wasn't expecting that, so have ended up here - where I have the cheap booking but am worried I'll get denied boarding on the day!