A really bad example, since they didn't tell you how much your life is worth to you.

It only changes what you'd pay proportionately, so it wouldn't make a difference.

No, because, as you say:

The real problem is that they didn't tell you how much you're capable of paying.

I implied the same ("pay ALL I HAVE (and try to borrow some)"), if maybe not as succinctly.

Your money is only valuable if you survive. Think of it as them reducing your winnings. It doesn't matter if you don't win. In that case, you should be willing to have them reduce it by $333 in either case.

all your remaining money magically disappears when you die.

Right, I ignored this last condition, which breaks the assumption of "your life is worth $1000" if you have more than that in your bank account. However, in that case there is no way to limit your bet, and the problem becomes meaningless:

If your utility function works like this, you can just abandon the finite part. It's effectively impossible for it to come up, and it's not really worth thinking about.

It's not mine, it's theirs (you lose everything you own, no matter how much). Which supports my point of a badly stated problem.