OP, here's another idea to ponder: On the application, you typically are asked to tell where else you are applying. Maybe having a short list signals to the AD that you are fairly serious about that school. Last year we did not do too many applications, and both DCs had more than one option.
I agree with others that it is foolish to apply when you are just lukewarm about a school, especially if you have a highly regarded public as a fallback.
The wait is almost over. Best of luck!