I believe both of your computations are correct, and the fallacy lies in mixing up the payoff for the group with the payoff for the individual - which the frame of the problem as posed does suggest, with multiple identities that are actually the same person. More precisely, the probabilities for the individual are 90/10 , but the probabilities for the groups are 50/50, and if you compute payoffs for the group (+$12/-$52), you need to use the group probabilities. (It would be different if the narrator ("I") offered the guinea pig ("you") the $12/$52 odds individually.)

byrnema looked at the result from the group viewpoint; you get the same result when you approach it from the individual viewpoint, if done correctly, as follows:

For a single person, the correct payoff is not $12 vs. -$52, but rather ($1 minus $6/18 to reimburse the reds, making $0.67) * 90% and ($1 minus $54/2 = -$26) * 10%, so each of the copies of the guinea pig is going to be out of pocket by 2/3* 0.9 + (-26) * 0.1 = 0.6 - 2.6 = -2, on average.

The fallacy of Eliezer's guinea pigs is that each of them thinks they get the $18 each time, which means that the 18 goes into his computation twice (squared) for their winnings (18 * 18/20). This is not a problem with antropic reasoning, but with statistics.

A distrustful individual would ask themselves, "what is the narrator getting out of it", and realize that the narrator will see the -$12 / + $52 outcome, not the guinea pig - and that to the narrator, the 50/50 probability applies. Don't mix them up!