Electric charge is Lorentz invariant (as shown in this question). Are weak isospin and color charge also Lorentz invariant quantities? My intuition says that they should be, but I've never seen any proof (rigorous or otherwise) to that effect.
If they are invariant, how can that be shown? Is there some deep reason why the charge of a relativistically consistent force must be Lorentz invariant?
Not quite but almost. There are some non-scalar Noether charges, such as
Of course, the first two only exist in Poincaré invariant theories, which we obviously assume to be the case. The third charge only exists in conformally invariant theories.
We now ask ourselves the following question: is there any other Noether charge that has one or more Lorentz indices? or are the rest necessarily scalars?
The answer is actually very interesting: under several rather general assumptions, we can actually prove that the answer is: all other Noether charges are necessarily scalars. This theorem is known as the Coleman-Mandula theorem, and you can find a very detailed and rigorous (up to physicists standards) proof in Weinberg's book on QFT (third volume). Some of the hypotheses of this theorem are:
The Noether charges are bosonic. If we drop this condition, we may have non-scalar charges, but the their form is constrained by a theorem due to Haag, Łopuszański and Sohnius (cf. here).
The theory has no massless particles (and therefore CFT's are out of the question). Whether this hypothesis can be lifted has been under investigation in the recent years, and as far as I know, no consensus has been reached (one could argue, what do we even mean by the $S$ matrix in a theory with massless particles?).
Scattering is non-trivial. Thus, we learn nothing about free theories from this theorem. Indeed, it is fairly easy to construct non-scalar Noether charges when the theory is free.
Some other technical (but natural) assumptions, which are not important here.
The take-home message is: under the hypotheses considered by Coleman and Mandula, all Noether charges are either $P^\mu,J^{\mu\nu}$ or Lorentz scalars -- they carry no Lorentz index and are therefore independent of the frame of reference.
In the spirit of the question cited in your question, you can think in the following way.
Suppose that the (classical) action is invariant under some fields and coordinates transformation. Corresponding Noether current $J^{\mu...}$is an object with the number of Lorentz indices determined by the transformation. For example, for the global and gauge transformations, which is the fields transformation, it carries 1 Lorentz index, for the Lorentz group transformations, which mix fields and spatial transformations, it carries 3 Lorentz indices, for the translation group transformations, which is coordinate transformation, it carries 2 indices.
Since the conserved current $J^{\mu...}$ satisfies the relation $$ \tag 1 \partial_{\mu}J^{\mu...} = 0, $$ one can construct the conserved lorentz-covariant charge $$ \tag 2 Q_{...} = \int \limits_{\Sigma} d\Sigma^{\mu}J_{\mu ...}, $$ where $\Sigma_{\mu}$ is the 4-hypersurface. Because of the conservation law $(1)$ it can be shown that $(2)$ is independent on the precise choice of the hypersurface $\Sigma_{\mu}$. Therefore we can choose time-like hypersurface, for which $$ Q_{...} = \int d^{3}\mathbf r J_{0...} $$ For your examples (corresponding to the internal $SU(N)$ symmetries) the current $J_{\mu...}$ carries one Lorentz index, and therefore the corresponding charges are Lorentz scalars.
