One issue I see right now (maybe Taedoju sees it too) is that of your ring position. Right now, it looks like the palms are facing backwards, with the rings forming a straight line. What you want is for the rings to be parallel to one another, with your palms facing each other (as though you were doing dips on parallel bars, not on a horizontal bar). What's happening now is that because of that position, your elbows must drift away from your body. It probably feels a bit awkward on your shoulders too. But most importantly, you can't go very deep because of that (you'd fall backwards if you tried to dip deeper) so your upper arms are barely clearing parallel.
Ideally, you change the grip to what I mentioned above, so you can dip deep. Like, hands-an-inch-away-from-your-shoulders deep. That way, you'll build the strength you need to come out from the bottom of the transition.
This video is very good:
If you build a deep, strong dip, and a good false grip high pull-up, then the transition takes a bit of skill work, but not too bad. Also, what is the exact reason for using weighted pull-ups? Would it not make more sense to use unweighted pull-ups, so that you can pull that much higher (sternum level) and build strength on a larger ROM? Just curious