I tried out adding more regularisation to the blue model (I just added a normal prior on the logit scale alpha parameters with most mass between averages 0.1 and 0.4) and adding an at-bats effect to the orange model. Code and a bit more writing is again here and here is the graph:
The at-bats effect removed the bias I mentioned before, but I think the intervals are still too narrow, especially for the low-data players. Thanks to extra prior the generalised Pareto model is no longer compatible with crazy true averages for low-data players.