I have a bit of experience with xray spectroscopy so thought I may be able to put in my two cents.
Composition matters for x-ray spectroscopy so knowing what acrylic is made from is important. A quick google search yields it is an organic compound of acrylonitrile which is composed of carbon, nitogen and hydrogen.
Since x-rays typically come from the innermost electron shells (k, l or m) of atoms it usually takes a large(ish) atom with a number of electrons to get a detecable x-ray to be emitted, although it can be picked up on light elements such as nitrogen and carbon (like we find in your acrylic) if the detector is sensitive enough and its window (point of entry for these high energy photons) is thin enough and of the right material.
The rule of thumb is the window must be made of a material at least 2-Z (two protons) less than that of the material being studied and is often beryllium (only four protons and hence four electrons) for this reason.
There are times we have seen a slight change in the photon emission energy of some samples due to the differing binding energy of some molecules and/or compounds, although we are talking about very slight and often at the limits of our detectors (a channel or two).
To determine if an acrylic was made via an extrusion or casting process would essentially mean trying to identify the difference in structure between the two processes, i.e. state of polymerization, crystallinity, defects, etc. Using xray spectroscopy to this end would be difficult if not impossbile without very sensitive low energy detectors and a library of sprectrums with which to compare them to. I am not saying they are not out there, but the paper you cited is emitting in the 30keV range, which is comparatively high.
The good news is there are better options as cast vs extruded acryclic has material properties that are much easier to resolve, such as elastic modulus, chemical resistivity, hardness, etc.; let's look at the latter.
If you need a non-destructive technique, then a simple hardness tester in the Shore range with an inexpensive durometer hand tester would easily be able to identify the difference between an extruded or cast acrylic sample. Here is an example of one I found on amazon:

These are simple and easy to use without the need for complex x-ray analysis tools that would likely struggle to do as well.
Hope this helps!