Is not it just true that for arbitrary partitions of these three submultisets onto parts $a_1+a_2=b_1+b_2=c_1+c_2=s$ at least 6 out of 8 combinations $a_i+b_j+c_k$ are not less than $s$?
If, say, the parts $a_1,b_1,c_1$ are large (at least $s/2$), then any combination with at least two large parts work, also either $a_1+b_2+c_2$ and $a_2+b_1+c_2$ works, say it is the former, and either $b_1+c_2+a_2$ and $c_1+b_2+a_2$ works.
Now just average.