I think one should be careful with the "fundamental-ness" of quantities as regards measurement vs properties. What we call a base quantity does not necessarily equate nor connote the quantity with a fundamental property. You yourself note that temperature is a complicated concept when put under scrutiny.
A physical quantity is something physical that can be measured in some way and that we then quantify. The choice of such units are arbitrary and tend to have a lot to do with history (although that's about to change). The list of four "properties" you have given are better phrased as base units and if you're talking base units, you have seven as given by the SI: metre, kilogramme, second, Ampere, Kelvin, mole and candela.
As units, they are a bit more "solid" to breaking down (and this allows us to perform useful things like dimensional analysis) but are by no means impervious; your system of base units can be anything you want, it merely has to be coherent and consistent. Of course, you'd have to convince the world to adopt your system as opposed to the one that's already in place to get anyone to work efficiently with you, but that's beside the point.
Charge is a fundamental property but not, I believe, in the way you've phrased it. The electromagnetic force is a fundamental property because it is one of the four fundamental forces: it appears irreducible to a more basic form or interaction. But it's not because of the base unit of charge. It arises from discrete symmetries, is additive, countable and is relativistically invariant. In fact, it's the other way around, the unit of charge is (will) be because of the elementary electric charge.
Thinking from the roots of units, what is charge ? "Amount of zap-zappiness", sure, but its unit is the Coulomb. What is a Coulomb ? Why, it's the amount of current transported by an Ampere in one second ! What's an Ampere ? An Ampere is flow of constant current such that, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10−7 newtons per metre of length.
Is then your charge a derivative of the metre and kilogramme ? Is then your candela merely a weighted summation of contributions of wavelengths of light ? Is then your kilogramme defining your newton, or should it be the other way around ? In special relativity, there's even a notion of expressing time in dimensions of distance, $ct$, the speed of light multiplied by time. Should we take that, then ?
Base units are base units, determined largely by consensus, convenience and usefulness. "Fundamentality" often comes from deeper arguments: symmetries, theories, equivalences, laws, etc. The "fundamentality" of a thing might inform the unit choice of something, but not vice versa.