Are there Planck units for weak or strong "charge", similar to the electromagnetic Planck charge $\sqrt{4~\pi~\epsilon_0~\hbar~c}~$?

All couplings in QFT are measured in Lorentz-Heaviside rationalized natural units.

That is, for instance, for the electric charge, $$ \alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} \approx 1/137 . $$ In these units $\epsilon_0=1$, so the elementary electric charge is simply $$ e = \sqrt{4\pi\alpha} ~\sqrt{ \hbar c} \approx 0.30282212 \sqrt{ \hbar c} \ . $$ The square root is called the Planck charge (in HEP; about a quarter of your definition above), $\sqrt{\hbar c}\approx 5.291 \times 10^{-19}$coulombs.

However, In natural units, one measures everything in units of $\hbar$ and $c$, and, e.g., GeV, a discretionary unit: the energy scale is left to itself, and is an inverse length scale, etc... Consequently, one sets $\hbar=c=1$, and the Planck charge is just 1, so e looks dimensionless in energy units---the only surviving dimension. And is about 1/3.

But you know that such charge units may be reinstated by dimensional analysis at the very end, uniquely, to produce a quantity to hand over to an engineer. In our case, if charge is really what you wish to hand over, you reinstate the above minuscule number in coulombs, (cruuuude mnemonic: recall the inverse Planck mass in GeVs).

Nevertheless, you'll probably never hand over a weak or strong charge to an engineer. In all likelihood, you'd reinstate $\sqrt{ \hbar c}$ in a rate or cross section, to make them dimensionally consistent. This is, in my book, the apotheosis of dimensional analysis. To sum up, the natural unit of charge is one.

For the electroweak interactions, you know that the above electric charge is $e= g \sin \theta_W = g' \cos \theta_W$, where g is the weak isospin coupling and g' the weak hypercharge coupling and $\theta_W$ is the weak mixing ("Weinberg") angle of about 28 degrees or so.

The strong coupling $g_s$ may likewise be inferred from experiment and is larger than the EW couplings at LHC energies, infinite at the confinement radius, and of the order of 1 in residual nuclear interactions... Think of the normalization of the Yukawa potential. I'd be shocked if you ever wished to measure it in coulombs.

To sum up, at the $M_Z$ scale, all these dimensionless SM couplings are, beyond e above: $g'\approx 0.357; ~ g\approx 0.652; ~ g_s\approx 1.221$. The last one, naturally, grows explosively with decreasing energy.

One other thing to note here:

We worked out a quantitative theory of electric charge in the 17th century. For the subsequent 250 odd years, we knew nothing about fundamental charge. So, a whole infrastructure was built up to describe charge, that then had to be retrofitted into the microscopic world. If we knew about atoms first, we surely would have just chosen $e$ as the fundamental electric charge (and probably something like a fraction of $N_{A}e$ for macroscopic amounts of charge), and built up everything from there, and been in gaussian units for electricity, and not even be aware of any coupling other than the fine structure coupling.

Well, this is precisely the case for the weak and strong forces -- we ONLY know of them at the microscopic level, and we never accumulate more than a few units of weak charge, and in principle, cannot accumulate even a single unit of net strong charge. So, we can happily describe these things using only fundamental units, and never have any reason to do otherwise.