There is no reason for considering the SI units as somehow fundamental to nature, but it is possible to explain why they are convenient for humans, so that an alien civilization will probably choose a similar convention of units, even though nature doesn't have units at the fundamental level.
The correct natural system of units for sure fixes $\hbar=c=1$, and $G$ to something order 1, determined by the details of the quatum gravity theory. So that there are no units at all in nature. The cgs system goes partway, and eliminates electrostatic units. This is perfectly fine, and cgs will work for any situation.
The reason for using units is that you have a kind of emergent scale invariance in certain regimes. This does not necessarily mean that the theories are exactly scale invariant, but that when you are given one theory with a certain scale, you can always imagine another theory with another scale also knocking around, and the two theories can talk to each other, and there is no reason to prefer one over another.
This type of scale invariance, scale indifference really, requires you to choose an arbitrary unit for convenience, to fix a scale. The fundamental theory is not scale indifferent at all, but we don't work with Planck-length sized black holes on a day-to-day basis, so we are in a scale indifferent regime. This justifies some of the SI unit choices.
For energies less than the Planck mass-energy, but everything relativistic, you have an approximate energy scale indifference, in the sense that there are particles of different masses that behave approximately the same. Choosing a mass scale in this regime becomes arbitrary, so you need a unit, traditionally the eV, for masses, and inverse length/times.
If you go nonrelativistic, time and space no longer scale together, and energies are quadratic in wavenumbers. Now for any fixed mass, you are allowed to scale space by a certain amount, so long as you scale time by the square of that amount. This nonrelativistic scaling law means that you to pick a unit of space, since c is no longer 1 but infinity, while the eV determines a unit of time because hbar is still 1.
If you go to macroscopic systems, the action of anything is enormous compared to hbar, and you lose your natural unit of mass. Then you can choose a unit of mass arbitrarily, and the eV trades in to a macroscopic unit of time, and the unit of length is still around.
In macroscopic systems, the dynamics is generally invariant under separate rescalings of mass, space, and time, in that if you have a system with certain values of any one of these, you can find rescaled systems which go twice as fast, or half as fast, etc. This is a rule of thumb, of course, magnetic effects coupled with electric effects determine c, and so relate time to space, but it is generally valid.
These three units are significant, because they correspond to the $G\rightarrow 0$ (low energy) $c\rightarrow 0$ (low speed) and $\hbar \rightarrow 0$ limits in which you pick up 3 arbitrary scales. The remaining SI units are not as significant, the mole and the Kelvin, the candela (this one is especially strange), and the Ampere (which is defined naturally anyway). The Kelvin can be replaced by a mole-Joule without any loss, the mole is really a pure number which describes how macroscopic your units of mass are in relation to atomic scales, and the Ampere is defined by the condition that two wires of 1 Ampere at a distance of 1m repel with a force of 2 10^{-7} Newtons, which allows you to define the Ampere as a certain combination of other SI units. The existence of approximate charge scaling can be used to justify a unit of charge, or equivalently a unit of current.