Is there a project that tracks and publishes data regarding weather prediction errors on a regular basis that would help with analyzing the correlation between incoming data and prediction error?
The World Meteorological Organization has several initiatives that track forecast performance. See for example here. The data is a bit delayed, as of today (Apr 5, 2020) the latest month available is February, which was before the strong decrease in AMDAR data.
For assessing forecast performance, one will need to find a forecast field for plotting and a certain metric. As forecast field I like to look at the 500 hPa geopotential height, since this is linked to the atmospheric flow, which itself largely determines surface weather. A suitable metric is the Anomaly Correlation Coefficient (ACC) which basically describes how well the predicted field correlates with the field as it was observed later on at the time for which the prediction was valid.
An intuitive way then to visualize these results, is to look at the forecast lead time at which the forecast reaches a certain threshold of ACC. In the example below this threshold is chosen to be 80%. So one could read the chart as 'how far the model was able to predict ahead', or in short 'predictability'.
The figure below (source: ECMWF) shows monthly means for this 'Day where ACC=80% is reached' over the Northern Hemisphere. Note the strong seasonal cycle with predictability being much higher in winter compared to summer for the Northern Hemisphere.
The large seasonality, strong year-to-year variability and in addition overall increases of forecast performance over time (with the model itself being improved over time as well!) will likely make it difficult to attribute any changes of the performance over time to the lower AMDAR coverage.
However, there have been several studies analyzing the relevance of different input data (including AMDAR) to the forecast models. I found for example this presentation. Although this is no conclusive answer to your question, I hope I provided some useful resources.