Going from the second moments of an object to its ellipticities and half-light-radius
You simply cannot obtain any information on the half-light radius from the quadrupole moment only of a galaxy, without knowing the galaxy profile.  Consider for example two (extreme) light profiles:


A galaxy consisting of a uniform disk of radius $R$ – that is, its surface brightness would be something like $I(r) \propto H(R_1 - r)$, where $H$ is the Heaviside function.
A galaxy consisting of a point source + a uniform disk – that is $I(r) \propto \delta(r) + H(R_2 - r)$.


You can easily find a combination of radii $R_1$ and $R_2$ such that the quadrupole moments of the two galaxies are identical.  However, clearly their half-light radii cannot be (for the second galaxy, in particular, the half-light radius vanishes).

From what I can see, the relation you wrote holds for a Gaussian profile, and even then it is not correct: your $r$ is not the half-light radius but the standard deviation of the Gaussian profile.  For different profiles there is no guarantee that your relation will work (in general it will not): in general, it will only give you something proportional to the square of the half-light radius, but the proportionality constant will depend on the specific light profile.