What about the following generalized cliquing approach?

Let E be the set of computable equivalence predicates between programs which are each a sufficient condition for functional equivalence (for instance, textual equivalence is in E).

For all e in E define the program
clique_e(other):
if e(self, other) then double down
else take winnings

On the first round, if you can prove that there exists e in E such that Quirrell's program is functionally equivalent to clique_e then double down, else take winnings.

This guarantees that you'll never self-destruct and you will get the maximum possible payoff as long as Quirrell remains cooperative.