Is there any way at all to handle iOS 10 notifications without having to upgrade to Xcode 8 and, possibly, have to rewrite whole swathes of code and re-instate all constraints (plus all the other scare stories)?
No, you are not able to do that. As with the new version of Xcode they will update the SDK also, so in case of Xcode 7 you don't have the updated SDKs.