Note that both sides are homogeneous of degree 4: ie, if we replace $a,b,c$ by $ta,tb,tc$ where $t>0$, both sides get multiplied by $t^4$ which has no effect on the inequality.
If we rescale by $t=1/(a+b+c)$, the effect is the same as requiring that $a+b+c=1$. Often this is formulated as the statement that we can choose $a+b+c=1$ without loss of generality. Alternatively, we could have chosen $a^2+b^2+c^2=1$ without loss of generality, which would be convenient for reformulating the LHS. Unfortunately, we cannot get both these assumptions at the same time. However, we can use then in a step-wise manner to break the inequality into three steps:
The way I got to this was the realisation that the RHS expression could be bounded from above more easily under the assumption $a+b+c=1$. Basically, if $a+b+c=1$, we have $abc\le1/3^3$, and so $RHS\le1/(2\cdot3^3)$. However, this inequality now only applies to $a+b+c=1$, so the return to the general case of any $a,b,c>0$, multiply by $(a+b+c)^4$, which is just $1$ when $a+b+c=1$ to make it
However, the outermost terms, $abc(a+b+c)/2\le(a+b+c)^4/(2\cdot3^3)$ is now homogeneous of degree 4 in $a,b,c$ and so holds for all $a,b,c>0$.
The LHS can be made easier to address if we assume $a^2+b^2+c^2=1$. Then, $a^6/(b^2+c^2)=a^6/(1-a^2)$. On this we can apply Jensen's inequality, but in terms of $u=a^2$, $v=b^2$, $w=c^2$ with $u+v+w=1$. Again, we work under the restriction $u+v+w=1$, and then multiply in the appropriate power of $u+v+w$ at the end to get homogeneous expressions of the same degree.
After the two simplifying bounds, we get the inequality in the middle of the topmost chain of inequalities, which shouldn't be too hard.