Can Boltzmann's Constant be Calculated from Fundamental Constants?

Simply by dimensional analysis, what you are asking for is not possible. $k_B$ is the fundamental unit of entropy, and no combination of other dimensionful universal constants can make one with units of entropy. A more rigorous treatment would use the Buckingham π theorem . Given the set of SI constants $c, G, \hbar, \frac1 {4 \pi \epsilon_0}, k_B$, we find that there are no dimensionless quantities that can be derived, and that every dimensionful quantity in SI can be expressed as a dimensionless quantity times some power of these units. That is the starting point in the definition of Planck units. The inclusion of $k_B$ in the set is crucial; the non-existence of any dimensionless quantities that can be constructed from these means that no expression of the form $k_B = f(c,G,\hbar, \frac1{4\pi \epsilon_0})$ can be found, because dividing the RHS by the LHS would result in such a dimensionless quantity.

What you could do would be to augment this set by one or more additional dimensionful constants $\kappa_i$, and decide that the $\kappa_i$ are more fundamental than $k_B$. One such example would be the temperature of the triple point of water (though few would claim this is "fundamental"). If you did this (and $\kappa_i$ has appropriate units) you could obtain a function $k_B = f(c,G,\hbar,\frac1{4 \pi \epsilon_0}, \kappa_i)$ which you could claim is a derivation of $k_B$ in terms of more fundamental constants, but really there is no such example of a quantity $\kappa _i$ which we would consider more fundamental than $k_B$ today, and any such function could be inverted to alternatively give $\kappa_i = g_i(c,G,\hbar,\frac1{4\pi \epsilon_0}, k_B)$ which we would think of as more "fundamental" (though ultimately what is "fundamental" is pretty arbitrary and mostly a matter of aesthetics).

One more thing, one could adopt an information-theoretic definition of entropy as in statistical mechanics. For example, $S = \log \Omega$, the logarithm of the number of microstates. If you adopt this convention then $k_B = 1$. But even still, there is nothing that stops us from instead defining $S = k_B \log \Omega$ and making entropy dimensionful. Or, even if you want a dimensionless entropy, you could make the logarithm base $b$ rather than a natural logarithm, in which case $k_B = \log b$.

Molecular dynamics can only very roughly estimate the triple point of water. The melting temperature is in principle the same. These authors claim an accuracy of 10 kelvin: http://www.pnas.org/content/early/2016/07/07/1602375113

One can doubt that. I would regard 50 kelvin already as impressive.

An ab initio definition of temperature or coldness could be $\beta = \frac{1}{kT} = \frac{1}{\Omega} \frac{d\Omega}{dE},$ the fractional change of the multiplicity with internal energy. Room temperature would then be approximately 4 % per meV for any system.

Anyway, in the impending redefinition of the kelvin, the value of the Boltzmann constant will be exactly fixed, and the triple-point temperature will be determined experimentally.