I've tried that driver but it didn't resolve the problem. I'm going to have a remote session with your support so hopefully you will understand the problem better seeing it live and not only via text or screenshots.
What is the distance to your devices? I have seen this issue with my Rii Mini Wireless Keyboard with Touchpad, which I use with the built-in BT in my NUC. Even though this device is rated for 10 meters, move it more than ~2 meters away and it hangs in funny ways - like as you have described.
BTW, I have the D54250WY. The keyboard came with its own BT USB dongle, but using it had worse problems - especially not working in BIOS setup after a cold boot.
The distance is about 10-15 cm. I've tried a lot of drivers sent to me from Intel Support and I even reinstalled Windows 10. The main problem now is that Windows cannot detect the integrated Bluetooth adapter even if the device is visible and said to be working in Device Manager.
Did you find a solution to the problem? Im having similar issues with a designer keyboard on my NUC5i7RYR (not H)
My keyboard will pair and connect, but then routinely drops out...
Sorry for a very late answer. I got my NUC replaced by Intel and everything worked as expected after that. So in my case it was hardware fault.
The support case took about 2 or 3 months from start to finish :-( Intel did some remote assistance from Costa Rica and England. I wasn't very happy with the help I got. I even bought a USB Bluetooth adapter and everything worked with that so I had "proof" stating the problem was hardware related but even if I showed them results, logs stating that the support nearly didn't believe me. But I didn't give up and after some more remote session, trying different version of the driver and showing that at third party USB Bluetooth adapter worked the NUC were sent to a service center for replacement.
Thank you for sharing the results, I am glad your computer is working again.