It’s definitely speed related. I landed at Biggin Hill again, before the 1000 ft marker, and taxi along its length at around 80kts. I didn’t hear ATC chime in until I got to the far end of runway 21, and started to apply brakes after I cross the threshold, and into the displaced runway section. I heard only a single call out that time, and immediately exited to the right.
So high speed taxi is one workaround! 
I didn’t keep an eye on my speed at the point ATC woke up, though.