In general, my own preference is to model the data according to (what I believe are) the characteristics of actual generative processes while also considering what your goals are. The toy data you were using previously had no upper bound at y=1, so the symmetric noise seemed fine. The skew normal noise seems to fit this new data quite well (to my eye). In contrast, the model with the symmetric normal noise looks quite poor. For example, credible values of y for x<500 look highly inconsistent with your data. In other words, your data looks very consistent with the “right angle” pattern you seem to be unhappy about. I’m not sure where your R^2 is coming from, but the eye-ball test would seem to prefer the skew normal model.
This works, but the results are really far shifted from my priors
This isn’t a problem on it’s own. Data should be allowed to shift your posterior from from your priors. If this isn’t possible, you can do without data.
and the mean function μ of the posterior sits pretty far out from the rest of the distribution.
The mean of the skew normal is not \mu, but rather \mu + \sigma \sqrt{\frac{2}{\pi}} \frac {\alpha }{{\sqrt {1+\alpha ^{2}}}}. Similarly, \sigma is not the variance. Details here.