I would first phone both airlines (both to ask about checking bags through, and to ask when check-in for the second flight opens).

OP can certainly try to see if the bags can be checked through to the ultimte desination, as long as the airlines have that agreement (e.g., arriving flight isn't on Southwest). They will, however, certainly need to check in for their international flight, show their passport, get their boarding pass, etc. Perhaps we're missing something about getting from arrival to departure gate, since taking a cab, offhand, makes no sense. If they can't check their bags through, and can't check in for the second flight right away, is there baggage storage at SEA? LAX has off-airport storage, run by a private company. Would allow OP to take light rail into downtown Seattle for a really quick visit.