I've been having a similar issue, although I lose all call audio both Bluetooth and earpiece. The microphone is also not working so nobody can hear me either. It's been driving me nuts!
So here's the rundown for all my details. I have the AT&T SGH-I537 on Firmware I537UCUCNE3 Android 4.4.2. I recently rolled the Baseband back to I537UCUAMF1 in hopes that it would fix it, but unfortunately it's still doing the same thing. However it does get much better 4G & LTE reception now. My Bluetooth device is a Jabra SP700 speakerphone kit. It worked great with the MF3 4.2.2 firmware and I even had some nice Tasker automation working with it, but I've disabled Tasker altogether in trying to track things down. I also recall something similar with the ML2 4.3 update, but not nearly as bad as this.
From what I can tell, if I make or receive a call over the Bluetooth kit, things are fine at first. Once I hang up everything goes to Hell. If I try to place a call, the phone makes a little blip sound and then it's just dead air. I can't hear anyone and they can't hear me. It's the same whether I'm connected to Bluetooth or just trying to use the earpiece. I think it also interferes with incoming calls because I've done tests with having people call me back after the call and it doesn't come in.
I have tried disabling Bluetooth, GPS, WiFi, and even going to Airplane Mode. I have disabled Tasker completely. I have avoided using Waze or any other navigation. So far the only thing that makes the phone usable again is to reboot it. I haven't yet tried messing around with the headphone jack. That may come next.
I just wish I could put ML3 4.2.2 back on it. Everything since then has been terrible. And now I'm out or warranty on the phone, so it's not like I can get it replaced. If I can get this resolved, it will last me until I replace it with a Project Ara device. Otherwise, I'm not sure what I'm going to do.
EDIT:
Well after a lot of testing, I ended up flashing stock NE3 back and now it's working again. So it must have been something that I did or had installed that caused the whole mess. So nevermind all that.