The lack of precision may be unnecessarily complicating things here. More precisely…
Wager 12.40830% on A and 0.33333% on B. The amount we should wager on the more favorable bet is greater than zero, but it is much smaller than the amount we should wager on A.
You’re correct that the push condition means Remark 1 cannot be applied directly. However the claim that no money is placed on the 1-1 draw at optimality was still incorrect.
OK. Now we’re in sync. The 0.333% bet is tiny but it isn’t zero. 
BTW, Lance, had you ever seen the Theorem that your wager should equal p times your bankroll where p is the probability of an outcome when you have the chance to bet (against an oddsmaker basing his payoffs on probabilities which sum to 100%) on all outcomes? It seems very elegant. Is it well-known?
That theorem is new to me, but the vast majority of the gambling math I have done professionally involves minimizing loss (either per ante or per total bet) in games where the house has an edge. Kelly betting and related concepts are merely a hobby of mine.