I’ve had tyres with damage to the rim protector like that passed on multiple occasions, sometimes with an advisory.
However, I don’t use Kwik-Fit, and they sell tyres for a living! So there may or may not be an issue.
As it was suggested above though, if they fail it for it, you’ll get a free retest so don’t be conned into paying over the odds just to have it back with a new MOT.