Is it possible to approximate given skewness and kurtosis values using an IFFT?
When you know that the process that generated the data is non-linear (and you are lucky to be in full control of the acquisition), you can try Attractor Reconstruction.

This technique attempts to reconstruct the trajectory a system may be taking through phase space, that results in the complex signal that is being recorded. The signal itself, may be showing chaotic behaviour with seemingly unpredictable spikes or a trace that looks like broadband noise. However, through Taken's theorem it is possible to find a "delay embedding" within which this behaviour is "explainable".

In practice, this looks like two (for example) pointers over the signal, separated by some distance (or delay) $k$. Given a one dimensional signal $x[n]$, this results in a two dimensional signal composed of $x[n], x[n+k]$ with $n,k \in \mathbb{N}, k&gt;0$. If $k$ (and the number of dimensions, here 2) is chosen correctly, the subsets of points (here $x[n], x[n+k]$) as plotted in a Cartesian plain, would reveal an attractor.

So, given this analyical framework, the task of "synthesizing a signal" becomes that of recovering the attractor, capturing it in a model and then playing it back at will, to generate the data.

As you may suspect, the choice of $k$ and number of dimensions is the main headache here. Theoretically, this should be equal to the fractal dimension of the attractor (which is not always known) but there are many different metrics and methods for estimating that.

An example of the whole process is available in this paper or, much more extensively in Kantz's book.

Finally, in more recent years, that "modeling" part of the attractor from above could not have escaped the Machine Learning treatment, so for the more up to date approach, you could refer to something like this one or even a GAN approach. These are everywhere these days.

You can see the...attraction here, especially when you are in control of the experimental conditions. I think that something like this would capture the dynamics you are after in a better way and manage to reproduce the conditions you are (were (?)) facing.

Hope this helps.