My situation is the same with the Surface Ergonomic Keyboard and my HP notebook. For a few months it worked fine then the phantom key repeat started and shortly thereafter the disconnect while remaining paired. Microsoft replaced the keyboard only to have it start up again shortly after receiving it. I wish I had seen this thread and this idea when this first happen.
Before seeing this post I downloaded the Bluetooth driver from HP for my model, ran the service pack but it only improved again for a short period of time.
Today, after reading this post. I decided to go to the source and get the driver from Intel. I dug around in system information and driver information to determine that my notebook has the Intel wireless module that includes Bluetooth, model AC 3168. While Intel showed the same driver version available for my model of Intel wireless module, Intel had different instructions.
The key seems to be in Device Manger to choose the Intel Wireless Bluetooth, then the driver tab and then to both uninstall the Bluetooth device AND the chose the optional Delete the driver software for this device before installing the new Intel software.