Alric,
I wanted to post an update to let you know a couple things I've found.
1) iOS 8 has had several BLE issues, and some specific to iPhone 6/6Plus, that are supposed to be fixed by iOS 8.1, of which a beta is reportedly currently being tested by developers and seems to resolve many issues (I have not tested this).
2) My issues seemed to have been mostly caused by using Tiles, and have been much improved by totally deleting the Tile App and removing their connections, in addition to doing a full reset of the Network (and thus Bluetooth) stack. Tile has come out with a public post discussing the issue (http://tiletogether.tumblr.com/post/98176762602/iphone-6-connectivity-updated-10 -5-2014), and due to the way they were architected, the issue can only be fixed by replacing the hardware altogether or waiting for iOS 8.1.
Were you able to sort out your issues? Did you remove all other Bluetooth devices and reset your Bluetooth stack?
Janssen