Is there a non-experimental reason for why $F=ma$ and not $F=qa$ is used in classical mechanics? Mass and charge are two fundamental properties of the universe. I hope you agree that they are different in that they denote different properties, but similar in that they are both fundamental. Space is, like mass and charge, a fundamental aspect of the universe. But space differs from matter and charge in that matter and charge are properties pertaining to physical objects, specifically the fundamental particles, whereas space is a property pertaining to the universe as a whole. Moreover, mass as a concept is not a priori, fundamentally interlinked with space in any way that charge is not. Both charge and mass apply to particles at a given location and time.
Is there then an armchair / non-experimental justification for why $F = ma = m\frac{dv}{dt}$ is chosen and not $F = qa = q\frac{dv}{dt}$?
An example of where this would have alternative prediction is about the acceleration of a charge:
$$
F = \frac{kq}{r^2} = q \frac{dv}{dt} \\
\frac{dv}{dt} = \frac{k}{r^2}
$$
As opposed to:
$$
F = \frac{kq}{r^2} = m \frac{dv}{dt} \\
\frac{dv}{dt} = \frac{kq}{m * r^2}
$$
And what would apply to charge acceleration in the usual mechanics would apply to gravitational acceleration in this mechanics.
Let's keep things classical, by the way.
 A: 
Is there then an armchair / non-experimental justification for why F = ma = m*(dv)/(dt) is chosen and not F = qa = q*(dv)/(dt)?

No, there is not. Newton's laws and Maxwell's equations have been verified experientially many times (in the regimes where they are valid). The reason we have the equations we do is because they describe reality as confirmed by experiment, not because someone just thought them up and nothing else was pursued further. One could definitely imagine a universe where $F=qa$; there is nothing wrong with this except that it doesn't agree with experiments.
Even you cannot throw out experiments, since you then ask about predictions

An example of where this would have alternative prediction is about the acceleration of a charge:

However, we already know these predictions would be false.

And what would apply to charge acceleration in the usual mechanics would apply to gravitational acceleration in this mechanics?

I suppose one could speculate about what this all would mean; however, this is not what this site is for.
A: See, physics is observational science. So my answer would inevitably invoke the observed phenomena.
So let's say we have defined a unit force and also defined the acceleration as the second order derivative of position (which requires unit length to be defined.), then we observe that for a given object which does not disintegrate during it's motion the force and acceleration both increase but their ratio remains constant. We call this constant the inertial mass of the object.
It turns out from equivalence principal that this inertial mass is equivalent to gravitational mass. Whereas we don't have any such principal which states the equivalence between charge and inertial mass.
