Is there a way to download the iOS 13 public beta IPSW?
No, it's not possible to download the IPSW files for public betas of iOS/iPadOS 13. You can, however, download the IPSW of the developer betas.

To do so, you should be a registered member of the Apple Developer program enrolled into a paid membership. Once you have the paid membership, visit the Apple Developer Download page(s) below to access and download the respective IPSWs:


https://developer.apple.com/download/#ios-restore-images-iphone-new
https://developer.apple.com/download/#ios-restore-images-ipad-new