I think that someone who believes in many-worlds will keep drawing cards until they die. Someone who believes in one world might not. An expected-utility maximizer would; but I'm uncomfortable about playing the lottery with the universe if it's the only one we've got.

Omega clearly has more than one universe up his sleeve. It doesn't take too many doublings of my utility function before a further double would require more entropy than is contained in this one. Just how many galaxies worth of matter perfectly optimised for my benefit do I really need?

The problem here is that is hard to imagine Omega actually being able to double utility. Doubling utility is hard. It really would be worth the risk of gambling indefinitely if Omega actually had the power to do what he promised. If it isn't, then you by definition have your utility function wrong. In fact, if exactly half of the cards killed you and the other half doubled utility it would still be worth gambling unless you assign exactly 0 utility to anything else in the universe in the case of your death.