Installing iOS 9 public beta (pb) 3 with iTunes
I would try again with the latest public beta - for many reasons in addition to being a more stable installation experience. 

If you have access to a developer certificate/account, the quickest way forward is to log in to developer.apple.com and download the beta build 13A4325c then point iTunes to it for the restore.

Since you have the other method to install the beta - using a signed profile to change the provisioning of the device OTA, you have a longer road ahead.


Wait it out (bad advice, but if you have time...)
Wipe the device after backing it up, place it in recovery mode and restore from iTunes. Then re-enroll in betas and hope you are on a fast network next download. Also, be sure the date/time are correct on the device before enrolling the profile.