I know things... A lot of things.
Join Date: Feb 2015
Motorcycle: GSXR 750 K4
Re: Wobling and damper problem
Although the video doesn't show the steering head, it does look like a true wobble alright. It's hard to tell from that point of view and I'm no expert in any case, but one thing is relatively clear. The instability started after landing from a small wheelie. It should be relatively safe to assume that you landed the front with the handlebars misaligned, that is, not looking straight, in the direction you're traveling. This has the same effect, as going straight at speed and yanking the bars to the side with all your might. (Actually, it's probably worse than doing that.) The wheel will try to realign with comparable force, it will overshoot and might start to wobble.
So you generally avoid that, but the steering damper is designed to help mitigate this problem, so either there is some problem with its operation, or the wobble was too big for it to handle (or at least to prevent entirely from happening, as the wobble was, happily, dampened in the end, one way or the other). There's also something else of note in the video. After landing from the wheelie at second :37, there's a period of a few seconds where there is a clear instability, which doesn't seem to develop, i.e. it doesn't seem to get worse. Then though, after about second :40, there's a very sudden (within less than a second) and great increase in terms of the violence of the shake. One explanation for that, since you mention starting in 2nd, but being in 4th when the problem developed, would be that you tried to shift gear at that point.
This would cut all power for a split second, hence causing a sudden drop in acceleration. If the front wheel was barely grazing the asphalt, due to the great level of acceleration up to then, it would then suddenly slam down onto the pavement, causing the latter to "grab" it and, if misaligned, yank it much more forcefully to the side, trying to straighten it again. That would be mistake number two, leading to 3-4 seconds of what looks like crying-for-my-mommy-level wobble. The fact that it straightened out in the end, might well imply that the stabilizer did its job pretty well.