I'm having the exact same problem with my Plantronics Pro Voyager HD headset as well as my cars built in bluetooth. Prior to upgrading my Nexus 4 to KitKat I could leave the Bluetooth on any device connected for hours if not days without issue and I could carry on conversations for hours at a time until the Bluetooth battery died.
At present the phone is completely unusable in the car, I have to keep pressing the headset connect/end call button and make it search for the phone in order to reconnect during the call.
There have also been strange instances where buzzing and clicking noises in the Bluetooth are heard in the middle of a phone call as I assume it loses the encryption key.
In fact Kitkat is so bad as far as the upgrade on the Nexus 4 that I'm thinking of rolling back to 4.3 just so I can get the Bluetooth working again.