Thanks.
Ugh, looks like a long slog between that Remote Terminal and TBIT.
I never looked for the airside shuttles at LAX. I think once I took the regular LAX bus and then after that, just walked as fast as I could, usually pushing the luggage cart, between terminals.
I also wonder about the connector between terminals 4 and 6. Is that behind the counters or something? It looked like the only way to get between terminals by walking was on the sidewalks. But that may have been more for DL and Skyteam flights, which operate out of different terminals.