(2018-01-01) Ahhh, so sweet, silent revenge donwvotes... Happy New Year!
Using the change of variables $x_1=a/b$, $x_2=b/c$, $x_3=c/a$, one asks for the maximum of $T$ under the constraint $S=0$, on the domain $D$ defined by $x_1\gt0$, $x_2\gt0$, $x_3\gt0$, where $ T=\sum\limits_{k=1}^3R(x_k),\qquad R(x)=x/(3+x^2),\qquad S=x_1x_2x_3-1. $ The extrema in $D$ are located at points such that the gradients of $T$ and $S$ are colinear. For every $k$, $\partial_kT=R'(x_k)$ with $R'(x)=(3-x^2)/(3+x^2)^2$ and $\partial_kS=(x_1x_2x_3)/x_k=1/x_k$, hence the condition is that $U(x_k)=x_kR'(x_k)$ does not depend on $k$.
The function $x\mapsto U(x)$ is smooth on $x\geqslant0$, increasing-then-decreasing and nonnegative on $0\leqslant x\leqslant\sqrt3$, and decreasing and negative on $x\gt\sqrt3$. Assume that $U(x_1)=U(x_2)=U(x_3)$ and call $v$ their common value. If $v\lt0$, the equation $U(x)=v$ has only one solution $x_v\gt1$ hence $x_1=x_2=x_3=x_v$ and $S\ne0$, which is absurd. If $v\gt0$, the equation $U(x)=v$ has at most two solutions in $(0,\sqrt3)$ hence either $x_1=x_2=x_3$, then their common value is $1$, or the $x_k$ are not all equal, then two of them are equal to some $x$ and the third one to $1/x^2$ and $U(x)=U(1/x^2)$. This last condition reads $W(x)=0$ with $ W(x)=(3-x^2)(1+3x^4)^2-x(3x^4-1)(x^2+3)^2, $ which has no solution $x\geqslant0$ except $x=1$. Finally, the gradients of $T$ and $S$ are colinear at the point $(1,1,1)$ and only there hence the only extremum on $D$ is $T(1,1,1)=3/4$, which is a local maximum since, for example, $T(x,1/x,1)\to1/4\lt3/4$ when $x\to0^+$.
To see what happens on the boundary of $D$, introduce the interval $K=[1/2,6]$. Then $R(x)\leqslant2/13$ for every $x$ not in $K$ and $R(x)\leqslant1/(2\sqrt3)$ for every $x\gt0$. Hence, as soon as one coordinate $x_k$ is not in $K$, $T\leqslant2/13+2\cdot1/(2\sqrt3)=0.731$. Since $0.731\lt3/4$, this proves that the supremum of $T$ is reached in $K\times K\times K$, and finally that this supremum is the maximum $T(1,1,1)=3/4$.
Caveat: The assertions above about the variations of the function $U$ and the roots of the polynomial $W$ were checked by inspecting W|A-drawn (parts of the) graphs of these two functions. To complete the proof, one should show them rigorously.