Called the AA RTW desk and am in the process of booking this DONE3:
BUD-DOH-CMB-BKK-HKG-NRT-BOS-ORD-YVR-ORD-BOS-MAD-TLV,AMM-LHR-BUD
(ORD both quick transit, DOH and LHR under 24h transit, TLV-AMM surface)


They said that they have to bill in the currency of the credit card's issuing country, so hopefully that means they'll convert the ex-BUD pricing to USD rather than quote me ex-USA pricing.

Thanks to everyone who helped!