Reminds me of the old medical maxim: if a problem has one solution, it works. If it has many, none of them work.
I've been battling with this issue intermittently for some weeks. Mouse stutter alone is extremely frustrating, but intermittently behaviour makes it even worse.
Like others, I found that turning Handover on/off, WiFi on/off, rebooting, remove the mouse and re-pairing, all seemed to "work", for a little while. Next time I reconnect - same issue appears and none of the previous "solutions" are effective.
But I think you might be on to the root cause: the DisplayPort!
I have a MacBook Pro (Retina, 13", Early 2015). There are two DisplayPort/Thunderbird ports near the MagSafe port on the left side. By default I plug an external monitor into the rear most, then turn on the mouse, and it stutters terribly.
After trying everything else, I tried unplugging the external monitor - bingo, no more stutter. I then tried plugging the monitor in to the other port. The mouse went stutter, stutter, perfect!
That's some pretty good evidence that all this Bluetooth mouse stutter drama might be related to other things plugged into the Thunderbird ports. Keen to hear other's experiences.