Well, there have been theories that derive the Fine Structure Constant from first principles, even though the theories they are based upon are not mainstream. Specifically, I am thinking of Laurent Nottale's Scale Relativity. Below is a 1992 paper on how it is estimated to within 1%. I believe newer research derives all of the coupling constants from first principles. Maybe this theory may give insight on where / how to calculate it in this framework.

https://www.bing.com/ck/a?!&&p=7aaa4a7437ffadbaJmltdHM9MTcyMjkwMjQwMCZpZ3VpZD0wNGIzZmU2MS0zMzllLTZhMGEtMThlMy1lZjAzMzJiMjZiNjImaW5zaWQ9NTIzMw&ptn=3&ver=2&hsh=3&fclid=04b3fe61-339e-6a0a-18e3-ef0332b26b62&psq=scale+relativity+fine+structure+constant&u=a1aHR0cHM6Ly9hcmlzdG90ZS5iaW9waHkuanVzc2lldS5mci9-bHV0aGllci9ub3R0YWxlL2FyYWxwaDkyLnBkZg&ntb=1

The U(1) gauge field should be obtained from permutations in the following way (quotation from here):

[...] for each vertex in a spatial hypergraph, there are many possible orientations in which a hypergraph replacement rule could be applied to that vertex [...], and we may interpret each such orientation as corresponding to a particular choice coordinate basis (i.e. some local section of a fiber bundle), which will subsequently place constraints on the set of possible orientations for other purely spacelike-separated rule applications. Thus, we can interpret the hypergraph itself as corresponding to some base space, with each vertex corresponding to a fiber [...]

This is a general idea, anyone is invited to develop explicit construction. Here is some progress in the direction of gauge theory in the Wolfram Model. I hope that, in the following Winter School, some people will work in this direction.

I have written a manuscript with some complementary ideas, mixing Eric Weinstein's framework with S. Wolfram's approach.

I will focus on the finite structure constant and let spinors for another discussion. First, let's see Seth Lloyd's characterization in context:

For example, the ordinary electrostatic interaction between two charged particles can be used to perform universal quantum logic operations between two quantum bits. A bit is registered by the presence or absence of a particle in a mode. The strength of the interaction between the particles, [...], determines the amount of time [...] it takes to perform a quantum logic operation such as a Controlled-NOT on the two particles. Interestingly, the time it takes to perform such an operation divided by the amount of time it takes to send a signal at the speed of light between the bits [...] is a universal constant [pi divided by 2 times the fine structure constant]

Ok, I agree that this characterization depends upon the electromagnetic interaction. In the Wolfram Model, electromagnetism can be developed as a particular case of gauge theory in the following way, I quote from the technical introduction:

In traditional physics, local gauge invariance already occurs in classical theories (such as electromagnetism), and it is notable that for us it appears to arise from considering multiway systems. Yet although multiway systems appear to be deeply connected to quantum mechanics, the aggregate symmetry phenomenon that leads to gauge theories in effect makes slightly different use of the structure of the multiway causal graph.

Seth Lloyd's characterization of the finite structure constant seems to be simpler than the subject of spinors. Indeed, it only depends upon the time it takes to perform [a quantum logic operation] and the time it takes to send a signal at the speed of light between the bits. In comparison with String Theory, Loop Quantum Gravity, and Geometric Unity, this characterization of the finite structure constant fits very well with the Wolfram Model, because it is computational. Furthermore, given any computer in the universe, it is possible to compute its finite structure constant using Seth Lloyd's characterization. The subtle part in the case of the Wolfram Model is that it is not a computer in the universe, but a computer generating the universe. So, time needs to be treated carefully in this definition. Concerning spinors in the Wolfram Model, at least 4 people (including myself) proposed their own versions (private communications). Here is my version (toy model).

Concerning

Seth relies on electromagnetism in his derivation [of the characterization of the finite structure constant]

I agree, but this fact does not prevent a non-electromagnetic derivation of the value of the finite structure constant. For example, the number pi is defined in geometry as the area of the circle of radius 1, but it is possible to find all the approximations of this number without involving circles by using the following program (exercise):

SquareFreePi[n_]:= Sqrt[6/N[#]]&[Mean[Boole[SquareFreeQ[#]]&/@Range[n]]]

For example, SquareFreePi[1000000] is approximately 3.1416 (good approximation).

Seth uses Coulomb interaction law between electrons. If he picked other carriers for information (say, neutrons), the constant would be different.

Nevertheless, the characterization

the time it takes to perform [a quantum logic operation] divided by the amount of time it takes to send a signal at the speed of light between the bits [...] is a universal constant [pi divided by (2 times the finite structure constant)]

does not involve any particle.

Thanks for the link of your spinor model. I looked though it, but I am not sure I can see how exactly it maps onto regular spinors.

Here is S. Wolfram's quotation related to my toy model of spinors (nevertheless, I do not see any need for spinors concerning the information-theoretic finite structure constant):

In the limit of sufficiently large hypergraphs, this shouldn’t make much difference, although it seems as if including directedness information may let us look at the analog of spinors, while the undirected case corresponds to ordinary vectors, which are what we’re more familiar with in terms of measuring distances.

Finally,

it would be nice to ensure that the structures you are getting in WM resemble regular particles (which are fermions), and fields (or bosons).

I would prefer pure information-theoretic bosons and fermions, rather than something resembling the actual particles, just to guarantee that the Wolfram Model will be a revolution in physics, rather than just evolution. For example, quantum mechanics is a revolution, it not the result of an evolution of Newtonian mechanics. Also, information-theoretic bosons and fermions could be useful for applications to computer science and social science (e.g., social laser).