I couldn't be bothered with any more than one change so I'd rather put up with clearing customs and immigration at MIA if it meant having one less flight. Availability via DFW is hard to get and JFK impossible or we would have gone there instead.
Would have chosen the LIM routing but sadly that's not an option any more so it looks like it'll have to be via MIA if we decide to go. Or we might just decide that Europe is a far easier and less stressful option altogether.

TAM are joining Oneworld in March next year so that might open up other possibilities.