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... clearly the most +EV move is to get Tex Barch out.

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This is wrong. Expected Value in the long run is that the jacks are the best hand and betting them will win the most.

There is no evidence that Hachem is a better player than Dannenmann.

What if ... has nothing to do with either players range of hands. I do not know what happened in the hand. Hachem is ahead against both players range.

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Tournament EV and Cash game EV are two different things. Usually they are the same, but in this case they are not. This is actually an interesting question because there is obviously a point where the "implied collusion" play goes from +EV to -EV (Tourney EV that is). This question is closer than I first thought. Out of the two "bad" outcomes that can arise for Hachem this hand.

1. Barch wins main pot (all three approx. even in chips)
2. Dannenman wins main pot, Barch eliminated. Hachem heads up down 2-1 in chips.
3. Hachem wins main pot, Barch eliminated (heads up with 2-1 chip lead).

The gap in tourney EV between outcomes 1 and 2 is MUCH greater than the gap between 2 and 3. So I believe the checkdown is the best play to maximize the chance that outcome 1 doesn't happen. If this were a cash game, outcome 1 would be no worse than outcome 2 so clearly betting would be the correct play. Also, like I said earlier, if there were two more players left, the chips would be more important than the player being eliminated.