Will drought help my turf woes?

My back yard is a combination of three grass varieties: St Augustine, Bermuda, and crab.

I know St Augustine doesn't handle drought well. I'm hoping that when this drought passes, the St Augustine will be killed off, leaving only the Bermuda behind (and I'm sure, still some crab grass to contend with).

Question: will it work? Is there anything I can do to help this process along?

If you hadn't guessed, I'm trying to get Bermuda established in the lawn.

Do y'all have any better advice on how to clean up my grass?

Thanks in advance!
Dave in Round Rock