I don't believe the first point, but I'm not entirely certain you're wrong, so if you think you have such a construction, I'd like to see it.

As for your second point, the number of times that Sleeping Beauty wakes up is always finite, so no matter what, the experiment does end. It's just that, due to the heavy tail of the distribution, the expected value is infinite (see also: St. Petersburg paradox). Of course, we would have to adjust rewards for inflation; also, the optimal strategy changes if the universe (or Sleeping Beauty) has a finite lifespan. So there's a few implementation problems here, yes.