Why wouldn't a simple expected value calculation work? You've shown that you can calculate the extra availability that subscribing to another region provides. Simply multiply the cost of an outage by the extra availability provided by an additional region that would have prevented that outage.

If expanding to another region costs more than just taking the outage, then it's categorically not a good option. If management still says no in the face of numbers that suggest yes, then that tells you that you're missing a hidden objection, and how you proceed will depend on a lot of factors specific to your situation.