Hmm. Your proof as written doesn't quite work because you have \\(\chi (a,b)\\) where it should be \\(\chi(a,b)\\), and there's only guaranteed to be a maximal hom object if \\(Obj \times Obj\\) is finite.
But I think both issues can be repaired:
Let \\(a, b\\) be objects of a \\(\mathbf {Cost^{op}}\\)-category. By the co-triangle inequality
\\[\mathcal{X}(a,b)+\mathcal{X}(b,a)≤\mathcal{X}(a,a) \\]
Adding a non-negative number can't decrease a real value, so
\\[\mathcal{X}(a,b)≤\mathcal{X}(a,a) ,\\]
however by the co- triangle inequality
\\[ \mathcal{X}(a,b)+\mathcal{X}(a,a)≤\mathcal{X}(a,b) \\]
by\\(\mathbf {Cost^{op}}\\) being a monoidal preorder
\\[ \mathcal{X}(a,b)+\mathcal{X}(a,b)≤\mathcal{X}(a,b) \\]
\\[ 2\mathcal{X}(a,b)≤\mathcal{X}(a,b) \\]
So \\(\mathcal{X}(a,b)= 0 \\) or \\(\infty \\).