Their video shows a pretty clear contravention whereby you enter the lane about 20 metre before it ends, presumably to turn left; true or false ? The video also shows other vehicles moving left at the end of the bus lane, unlike you who would seem to want to get some advantage over them. That's no me thinking, it will be the thinking of an adjudicator at London Tribunals if you take them there.
You show their letter of rejection but not your reps, which we really do need to see to check if they are bluffing their way or not.