Trying to book a complicated multi-city ticket:

LAX > Tokyo > Taipei > LAX. 2 Adults 2 Children (6 and 8), however, in Taipei, we are meeting up with their grandparents who are taking the kids on a longer vacation and then all 4 of them will fly back later.

Trying to find the cheapest route and I can get it $5XXX if I book 2 adults for the full route and then two children for just LAX > Tokyo > Taipei. However, I see I have to call in to get around the unaccompanied minor fee.

Tried booking over the phone to avoid the unaccompanied minor fee but the rep said they can't link those two proposed reservations together because they're not the same routes. They try booking 2 adults and 2 children for LAX > Tokyo > Taipei and then a separate ticket for 2A from Taipei to LAX but the price is much higher, almost $7XXX.

The rep said they couldn't do it the way I proposed and after asking for a supervisor I get stuck in phone limbo and lose the call.

Is my first option possible? Am I asking the wrong things? Would love advice on how to book this for the price I'm seeing without having to pay an unaccompanied minor fee. I've read that some airlines charge the fee at the airport so I would be able to show that adults are traveling with them on their whole route and maybe get around it then?

TIA