If all that Omega offers is a series of bet, and the TDT agent has no information about what would have Omega done if the coin toss came out differently or even if he will see Omega again, then it's not clear to me what a TDT agent should do.

Indeed. I think that's why it's sometimes better to imagine "Omega" as some sort of stable physical process whose complete functionality we know, instead of an as an agent with mysterious motivations.