Should I somehow try to assess my likelihood of meeting Omega (in some form or other) and guess what sort of parlour game it is likely to play with me, and for what stakes? Has anyone got any idea how to do that assessment, without unduly privileging the games that we happen to have thought of so far? This way madness lies I fear...

Not exactly madness, but Pascal's wager. If you haven't seen any evidence of Omega existing by now, nor any theory behind how predictions such as his could be possible, and word of his parlour game preferences has not reached you, then chances are that he is so unlikely in this universe that he is in the same category as Pascal's wager.