OK, your video shows the roadworks, but you would seem not to have captured the "Use Bus Lane" sign.
Their letter is the usual Fob-Off letter that totally fails to consider your representations. TfL have, like all authorities serving PCNs a duty to consider representations but here, although repeating most of yours, then totally fails to respons to them.
If it were me, I'd take them all the way to London Tribunals as your evidence seems very strong, but sight of the sign would make a slam-dunk win, I think.
So over to you