Only on FT would you have somebody going from FRA-BKK via ORD and TYO. And then desire to switch to EK from ORD and backtrack to DXB to BKK. You sir, are my hero
The real question is where are you going after BKK and when. I'm scared for you (and any companions traveling with you) if you're going to do FRA-ORD-HND-BKK-??? without any real stops in between, even in F. If you misconnect in ORD, you can push to take the next day NH via NRT. I never seem to have a problem finding FRA-ORD in LH F when I'm looking for FRA-LAX. Start lookin at 14 days out, and keep checking until departure.
Another obvious answer is skip ORD all together and go through IAH/SFO/LAX but that would require you find NH F space again. So finding FRA-ORD in F is your best bet.