mid-to-late October is a very good time for
Fiji plus there's a convenient nonstop flight from Nadi to LAX since I'm being realistic about End City.
on the topic of 'being realistic' I can't see them skipping China again so perhaps something like
legs 7, 8 - Harare or thereabouts, Zimbabwe
leg 9 - China (Xi'an maybe
)
leg 10 - train ride to somewhere else, China
leg 11 - Fiji
leg 12 - LA or somewhere nearby
Just by checking the number of options, Taiwan (and Australia for that matter) seem like a better fit for the start of the race, where they can easily get 11 teams on two flights but I wouldn't be surprised to see either.