But we end up having to make other arbitrary assumptions about what happened at the very beginning of the big bang or about multiverses that we have no scientific justification in making. Calling the observed physical constants 'arbitrary' is a very mathemetician-like move. It is true that they are free variables in the sense that they cannot currently be derived--mathematical models of the universe must include these constants as hard-coded values. But the word arbitrary implies that they could have been something else, and we really don't have justification for that.
We write the modeling program and somewhere we write
FINE_STRUCTURE_CONSTANT=1/137;
It is very tempting to instead write
./my_universe.sh --fst=0.007 ...
The code does the same thing, but as far as we know that constant is as invariable and intrinsic to what it means to be a universe as PI is to what it means to be a circle.
It is fun to think about but it can't produce scientific results. It's epicycles, coming up with more complicated stories that over-fit for a singular observation.