Since nobody answered, I will give it a try.
One IMHO good way to interpret more complex models is via their predictions. You could for example use the model to plot counterfactual predictions - how the model would expect the outcome to change for each individual actual project when varying the weight function. For example, in the figure below, each line connects counterfactual predictions (vertical axis) from one posterior sample for a single individual as I change a single categorical predictor (horizontal axis), other predictors stay the same as in the original data:
The vertical bars are then 50 and 95% quantiles of the predictions. The color indicates whether the outcome has risen or decreased between the neighbouring levels of the categorical variables and the green points (which are a bit of a clutter) show average outcomes for individuals with the given predictor level across the original dataset.
Some similar notes in my earlier post:
Alternative approach is to predict for a new, unseen project (drawn from the same population of projects) - you just make the data for prediction include a new project ID/name and set allow_new_levels = TRUE. Then you can predict for various weighting schemes and LOC and summarise - this gives you a grid of estimates.
Finally a minor correction (but maybe you are already aware of this, so sorry if this is obvious):
Not really. The plot is consistent with a difference of ~0.3 - this might be non-negligible difference for many applications so saying there is no difference is not warranted. Maybe the parameters are highly correlated and if you plot the posterior of the differences it is more narrow, but this plot does not warrant the conclusion you stated. (later you show that the model is not consistent with a difference larger than 0.015 on the outcome scale which might plausibly be negligible).
Hope that helps.