"a karma doctrine of reward / improvement"

I think you are misunderstanding the doctrine of karma. All that karma implies is that future lives are determined by behaviors in previous lives. It does not imply any sort of progress or growth in virtue. In the Buddhist view, in particular, there is no God, nor any being doling out rewards and punishments. Karmic results are not really rewards and punishments proper but natural consequences of past actions. A good rebirth is simply the result of good actions in a past life, while a bad rebirth results from bad behavior at some point. There is nothing in the doctrine indicating any sort of progress towards the good.

You're assertion that karma doctrines would imply a cumulative improvement over time is incorrect. It does not imply that. If I am good in this life, my next rebirth will be a good one. You're argument in the note in Metaphysics by Default is based on the assumption that "good rebirth" means virtuous future life. It doesn't. If I am exceptionally good in this life, I may be reborn as a billionaire, but billionaire's are seldom ethical people, so my next rebirth after that is likely to be a bad one. And so there is no forward trajectory or progress.