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Theoretically, you could ask the gate agent to change the number on the ticket prior to departure from DFW in each direction. Anecdotal reports at FlyerTalk and elsewhere, however, suggest this does not really work in practice, either because of limitations in the reservations system, or because the gate agents may not be familiar with the exact way to record it in the reservations system. They are, after all, extremely busy prior to departure.
What you could do instead is have no frequent flyer number on the reservation at all and claim the flight credit retroactively using the ticket number from the airlines' respective websites.
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Generally, you can only earn mileage with one program. For SkyTeam, I am pretty sure you won't be able to earn mileage twice for the same flight sectors. You need to provide ONE membership card, and that's that. I tried, a few years ago, with AF and KE, and I was told, sternly, that I had to pick one. I chose AF.
When I later on tried to request mileage adjustment with KE, I was asked for my boarding passes, which showed clearly my AF membership. Whoopsies.
Well, you said
KE earns 12K+ on ICN-DFW-ICN alone, AA will earn about 10K award miles for the whole trip
So obviously I assumed you were trying to double-dip :-) If DFW<–>EZE is not eligible for mileage on KE, you could claim with AA. On the other hand, if all segments are eligible, you might get locked in, especially if it's a single ticket.
I assume you'll have to pass border control and customs in DFW, as is customary in the US. Are you sure your luggage will be checked in all the way though? You might have to go to an AA counter anyway, and it would be the right time to claim mileage for the DFW-EZE segment.
On the way back, depending on whether you get both boarding passes in EZE, or have to get DFW-ICN in DFW, you could show both cards, one for each segment. Moreover, the airlines will have by then your membership numbers.