So you've submitted the application and answered yes? Therefore, unless it can be withdrawn you'll most likely be denied.
If you end up with an FID denial on your record you'll have pay the $750 anyway then reapply to get it cleared
Sorry, but if you've found a Dr. that is willing to write a letter satisfying the Middletown PD just fork over and be done with it.
Others have paid much more for expungements in order to secure an FID.