I'm having problem with a weird but cheap international routing I found.

It's SIN -> LHR -> ORD -> DTW. However, for the SIN -> LHR leg it is covered by British Airways and the LHR -> ORD -> DTW legs it is covered by American Airlines. Thus, I can't seem to book this routing through the carrier website:

British Airways won't acknowledge this routing on their website and will select a more expensive "all BA" route.

American Airlines acknowledges this routing, but it does not have a local (SIN-based) website, so the calculated price is higher. Yes... if you put Singapore (SIN) as the starting city the price is like 30% lower than if you use an American city and purchase the fare in USD.

Can anyone help me with this? The best I've found is booking through Zuji, but they charge a full $100 for just booking the fare. I guess I'm used to the base price off the ITA website through the carrier websites...