I used to have the same problem, or close enough - I'd be able to do one or two from a dead hang, but then I'd have to cut the ROM very slightly at the bottom in order to keep doing pullups.
I solved this problem for myself by doing pullup shrugs. I'd hang at the bottom (not slack, but at a dead hang). I'd keep my shoulders tight and "shrug" upwards 10-15 times at the end of every pullup workout. So maybe I'd do 3-5 sets of pullups avoiding the "dead hang" at the bottom, then do a set of those shrugs. Each shrug was just a straight-arm pull, tightening up the lats and shoulders.
Now dead hang isn't a particular problem even for repeated sets or long sets.
I'd try that. Can't hurt, since it's just a set of very short range motions with a lot less stress than coming down hard from a kip.