Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory.
How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$?
Really, even pointers on how to say anything meaningful about these $p$ are welcome. Originally I also asked about how to count the density of all $p$ (not just those of the form $3k+1$) such that $2$ (or $3$) is not a cubic residue modulo $p$, but Felipe Voloch's comment quickly addresses how to deal with them, via Chebotarev's density theorem.
The difference between the question and these easier problems is that here I am asking that $k+1$ is not a cube modulo the prime $3k+1$, so the same approach does not seem to apply.
Finally, if it turns out that the density is not zero, how does one go about finding the density of those $p=3k+1$ that satisfy that none of the equations $x^3=2$, $x^3=3$, $x^3=k+1$ have solutions?
(Ideally, the techniques lift to other situations, such as studying fifth powers modulo primes $p=5k+1$, etc, but even methods exclusive to the case of cubes are very welcome.)