Statistical query model with Gaussian noise?
We know something close to what you want.

If you look at Ke Yang's "Honest Statistical Queries" -- there is no noise at all, but only "sampling error".   In this model, you pass in a parameter $t$, and the Oracle takes $t$ samples, honestly evaluates the passed-in function (onto {0,1}), and returns the average value of the function on the samples.

In effect, this estimates the expected value of the function.  It is easy to see that this Oracle samples from the Binomial distribution $B(t, E_{x \sim D}(f(x))$.  For small skew and large $t$, this is approximated by the Normal.  We extended this to explicitly consider variance here.

The punchline, basically, is that SQ lower bounds still hold in this non-adversarial noise setting.  Actually the SQ lower bounds are even harsher because they work for sample sizes $t=1$, where the algorithm sees the result of its function evaluation on every random sample (and can average or do anything by itself).