A few results that addresses, but not quite answers my answer:

  - In the seminal paper of Gidas-Ni-Nirenberg, it is mentioned that, at least in the bounded domain case, that a positive solution to $-\triangle u = g(u)$ need not be radially symmetric when $g$ is not at least Lipschitz continuous. In particular, they gave a family of solutions and equations for which $g \in C^{0,1-2/p}$ for any $p > 2$. 
  - Along the same veins, it is not too hard to construct a counterexample to Gidas-Ni-Nirenberg if $g$ is merely continuous. Let $u(x) = v(x-x_v) + w(x- x_w)$, where $v$ is a radially symmetric bump function supported in the unit ball and monotonic radially, and $w$ is a radially symmetric function such that $w = 1$ in the ball of radius 5 and monotonically decreasing to 0. Then one can define $g$ by considering the case $x_v = x_w = 0$, which by a bit of undergraduate analysis one can show must be continuous. But the local nature of the equation means that for $|x_v - x_w| < 2$, the same equation will still be satisfied. Note that if one can construct one such example for which $u$ is actually the minimal action solution, then this will give a bona fide counterexample.  
  - [Franchi, Lanconelli, and Serrin](http://www.ams.org/mathscinet-getitem?mr=1378680) shows an existence theorem for radially symmetric, non-negative solutions to a large class of PDEs, which includes $-\triangle u = g(u)$ for $g$ merely continuous. The corresponding uniqueness theorem they proved in the paper, however, is only concerning "uniqueness among radially symmetric solutions" and in particular does not use the variational structure at all. In particular, this result needs to be consider together with [that of Cortazar, Elgueta, and Felmer](http://www.ams.org/mathscinet-getitem?mr=1364001) which shows that for a particular equation of the form above with a manifestly non-Lipschitz nonlinearity, the equation admits compactly supported nonnegative solutions. This gives yet another counterexample to Gidas-Ni-Nirenberg in the low regularity case. 


Neither of the above uses the variational structure of the  "ground state". One result which I found that does use the variational ground state is that of 

 - [Flucher and Muller](http://www.ams.org/mathscinet-getitem?mr=1617704), which was able to show that under the assumption that, roughly speaking $g(0) = g'(0) = 0$ and that the corresponding $G$ is non-negative, all variational ground states agree _up to a translation outside a compact set_, and that if $g$ itself is non-negative, non-negative ground states are in fact positive and strictly decreasing radially, hence unique (by the result of Brothers and Ziemer mentioned in the question). So in the general case they were not able to rule out the possible counterexample given in the second bullet point above.