That makes no sense in context, since continuity is equivalent to saying (roughly) 'If you prefer staying on this side of the street to dying, but prefer something on the other side of the street to staying here, there exists some probability of death which is small enough to make you prefer crossing the street.'

This sounds almost exactly like what Houshalter is arguing in the great-grandparent ("these events have such low probability that I can safely choose to ignore them,") so it can't be the axiom s/he objects to.

I could see objecting to Completeness, since in fact our preferences may be ill-defined for some choices. I don't know if rejecting this axiom could produce the desired result in Pascal's Mugging, though, and I'd half expect it to cause all sorts of trouble elsewhere.