Guaranteed.. You submit a Delaware CCDW Application without providing your SSN it will be denied. This will not change until it is removed from the application which will require a lawsuit.
Keep your PA residency, and your Utah permit will allow you to carry while in DE. If you wish to not submit your SSN then apply, wait for it to be denied and file a lawsuit like Michael Stollenwerk did in PA and see where it goes over the following couple years. Use Facebook, KickStarter, local De associations, etc to gain some funds for your cause to assist in any costs, if needed. Your Utah permit will see you through. If you win you can change your residency and apply right away as you'll no doubt have the references by that time.
While you're at it, file a lawsuit to allow a new resident who has a permit from another state we share reciprocity with to continue using it until the Delaware application has been either approved or denied.
If you win you'll be happy and many Delaware residents will applaud your actions. If you loose you can still be happy that you stood up for your ideals and many DE residents will still applaud your actions.